I realize that this is likely to depend on jurisdiction (EASA or FAA would interest me the most in that case), but is there any common-sense kind of line that you don't cross until you've at least checked that it's okay?
I mean it's obviously okay to put checklists in a side-pocket, and it's obviously not okay to replace the avionics. Can I replace the PTT button on the yoke? Or if I don't touch the wiring and simply add some plastic on top give it a better feel?
This answer applies to FAA certified aircraft. For experimentals, there are really no restrictions. I mean, you can build the airplane from scratch so replace what you want... but for certified aircraft:
The following is copied and pasted from the AOPA website:
FAR Part 43 specifies who may do what to an aircraft in the way of maintenance, repair or alteration. It requires that only properly certified mechanics work on aircraft and "okay" them for return to service. However, it does allow preventive maintenance to be performed by a certificated pilot, holding at least a Private certificate, on an aircraft owned or operated by that pilot, provided the aircraft is not used in commercial service. The responsibilities for a pilot performing preventive maintenance are very similar to those imposed on the certificated mechanic performing other duties. The FARs require that anyone who works on an aircraft must have the appropriate maintenance and service information available. This means quite simply that before you set about performing preventive maintenance items on your airplane, you must first have the proper maintenance manuals available.
A list of "preventive maintenance" allowed is specifically listed in FAR 43 appendix A (4) c and is copied below:
Preventive maintenance. Preventive maintenance is limited to the following work, provided it does not involve complex assembly operations:
(1) Removal, installation, and repair of landing gear tires.
(2) Replacing elastic shock absorber cords on landing gear.
(3) Servicing landing gear shock struts by adding oil, air, or both.
(4) Servicing landing gear wheel bearings, such as cleaning and greasing.
(5) Replacing defective safety wiring or cotter keys.
(6) Lubrication not requiring disassembly other than removal of nonstructural items such as cover plates, cowlings, and fairings.
(7) Making simple fabric patches not requiring rib stitching or the removal of structural parts or control surfaces. In the case of balloons, the making of small fabric repairs to envelopes (as defined in, and in accordance with, the balloon manufacturers' instructions) not requiring load tape repair or replacement.
(8) Replenishing hydraulic fluid in the hydraulic reservoir.
(9) Refinishing decorative coating of fuselage, balloon baskets, wings tail group surfaces (excluding balanced control surfaces), fairings, cowlings, landing gear, cabin, or cockpit interior when removal or disassembly of any primary structure or operating system is not required.
(10) Applying preservative or protective material to components where no disassembly of any primary structure or operating system is involved and where such coating is not prohibited or is not contrary to good practices.
(11) Repairing upholstery and decorative furnishings of the cabin, cockpit, or balloon basket interior when the repairing does not require disassembly of any primary structure or operating system or interfere with an operating system or affect the primary structure of the aircraft.
(12) Making small simple repairs to fairings, nonstructural cover plates, cowlings, and small patches and reinforcements not changing the contour so as to interfere with proper air flow.
(13) Replacing side windows where that work does not interfere with the structure or any operating system such as controls, electrical equipment, etc.
(14) Replacing safety belts.
(15) Replacing seats or seat parts with replacement parts approved for the aircraft, not involving disassembly of any primary structure or operating system.
(16) Trouble shooting and repairing broken circuits in landing light wiring circuits.
(17) Replacing bulbs, reflectors, and lenses of position and landing lights.
(18) Replacing wheels and skis where no weight and balance computation is involved.
(19) Replacing any cowling not requiring removal of the propeller or disconnection of flight controls.
(20) Replacing or cleaning spark plugs and setting of spark plug gap clearance.
(21) Replacing any hose connection except hydraulic connections.
(22) Replacing prefabricated fuel lines.
(23) Cleaning or replacing fuel and oil strainers or filter elements.
(24) Replacing and servicing batteries.
(25) Cleaning of balloon burner pilot and main nozzles in accordance with the balloon manufacturer's instructions.
(26) Replacement or adjustment of nonstructural standard fasteners incidental to operations.
(27) The interchange of balloon baskets and burners on envelopes when the basket or burner is designated as interchangeable in the balloon type certificate data and the baskets and burners are specifically designed for quick removal and installation.
(28) The installations of anti-misfueling devices to reduce the diameter of fuel tank filler openings provided the specific device has been made a part of the aircraft type certificate data by the aircraft manufacturer, the aircraft manufacturer has provided FAA-approved instructions for installation of the specific device, and installation does not involve the disassembly of the existing tank filler opening.
(29) Removing, checking, and replacing magnetic chip detectors.
(30) The inspection and maintenance tasks prescribed and specifically identified as preventive maintenance in a primary category aircraft type certificate or supplemental type certificate holder's approved special inspection and preventive maintenance program when accomplished on a primary category aircraft provided: (i) They are performed by the holder of at least a private pilot certificate issued under part 61 who is the registered owner (including co-owners) of the affected aircraft and who holds a certificate of competency for the affected aircraft (1) issued by a school approved under Sec. 147.21(e) of this chapter; (2) issued by the holder of the production certificate for that primary category aircraft that has a special training program approved under Sec. 21.24 of this subchapter; or (3) issued by another entity that has a course approved by the Administrator; and (ii) The inspections and maintenance tasks are performed in accordance with instructions contained by the special inspection and preventive maintenance program approved as part of the aircraft's type design or supplemental type design.
(31) Removing and replacing self-contained, front instrument panel-mounted navigation and communication devices that employ tray-mounted connectors that connect the unit when the unit is installed into the instrument panel, (excluding automatic flight control systems, transponders, and microwave frequency distance measuring equipment (DME)). The approved unit must be designed to be readily and repeatedly removed and replaced, and pertinent instructions must be provided. Prior to the unit's intended use, and operational check must be performed in accordance with the applicable sections of part 91 of this chapter.
(32) Updating self-contained, front instrument panel-mounted Air Traffic Control (ATC) navigational software data bases (excluding those of automatic flight control systems, transponders, and microwave frequency distance measuring equipment (DME)) provided no disassembly of the unit is required and pertinent instructions are provided. Prior to the unit's intended use, an operational check must be performed in accordance with applicable sections of part 91 of this chapter.
My answer below is largely geared toward FAA regulations and US-registered aircraft.
Other areas may have different governing regulations (e.g. for Europe it's EC 2042/2003 - Appendix VIII), but there seem to be some commonly-accepted things pilot-owners may do on their aircraft.
Broadly, on certificated aircraft pilot-owners are typically permitted to make minor repairs to aircraft they own and operate non-commercially. For example you can
...provided "complex disassembly" is not required to complete the above tasks.
It is also generally accepted that pilots are allowed to add fuel (duh), engine oil (again, duh), water/coolant (for liquid-cooled engines), and to replenish hydraulic fluid in accessible reservoirs that can be serviced without opening the system to air & requiring it to be bled.
The US FARs actually allow a great deal more than that (see Skip's answer which includes the relevant section of the FARs), the replacement of certain types of tray-mounted radios, some fabric repair (on fabric-covered aircraft & balloons), and repainting (with the exception of "balanced control surfaces"), however much of that starts getting into the kind of work where special equipment or facilities would be extremely helpful (if not necessary) to ensure the job is done properly.
There are some exceptions to the above (this is by no means an exhaustive list of exceptions):
