Is there a maximum or minimum temperature that a plane can take off or land at?
There is a maximum temp but that varies by airport altitude, runway length, aircraft and payload. You can find an answer to that question here.
There is not really a minimum temperature but it needs to be warm enough out to run the engines. Again this varies by aircraft and a preheat can usually solve this problem but there are places on earth where it be comes practically to cold to operate some aircraft. On a similar note if icing conditions prevail you are not really going anywhere but those can occur at lots of altitudes and temperatures. If the given airplane is stored outside and the airport lacks proper de-ice equipment you may be grounded.
Maximum and minimum temperatures are governed by two factors.
First, there is the limitations imposed by the manufacturer. For example, recently the media has been reporting that some regional jets have max temps for operations, and they have been unable to operate in the southwest US. Some manufacturers also limit low temperature operations, and may require different lubricants, fuel, etc.
Second, aside from manufacturer (or company) limitations, there are practical limitations. Engines can be hard to start at -40. Fuel can gel. Batteries can render insufficient power to perform normal starts. These limitations are normally determined by practicality, experience or other sources of information by the crew.
In my experience, dealing with hot soaked planes is normally easier than dealing with cold soaked planes. Other issues like density altitude (DENALT) are operational issues, that while normally are considerations, are usually not hard limitations. For example, departing a 1200 foot strip in South American mountains on 35C days has been a limitation with a turboprop aircraft, for me, but the number of times I have had those operational considerations limit flight can probably be counted on my fingers and toes.
So in general, there are two limitations: Those imposed by the manufacturer and the aircraft certification, and those imposed by practicality (which includes best practices).
Yes, there are some temperature limitations, but these are typically aircraft specific.
For example, the Beechcraft King Air B200 has the following limitation:
Max Outside Air Temperature Limitations
Sea Level to 25,000 FT pressure altitude: ISA + 37° C
These limitations are given in the AFM, and compliance is required.
Practically speaking, this limits aircraft operation of any kind—including takeoff and landing. Takeoff would not be authorized at sea level above 52°C, or at 5000 ft pressure altitude above 42°C.
For a minimum temperature, the engines have a starting limitation of at or above -40°C.
It depends on the model of the plane, particularly its engines, the length of runway available, and surface wind conditions. The aircraft's Pilot Operating Handbook will include a set of performance charts. To determine whether you can safely takeoff or land, you would plot the values (including temperature) into the chart.
This related question discusses some of the precautionary steps one should take when operating in very cold or hot weather.
In general, aircraft performance will be the limiting factor in hot weather, and in cold weather the challenge is to get things started and stop things from freezing.
There is no absolute temperature regulation-wise. Practically you can argue that the minimum temperature is when the fuel freezes and the maximum temperature is when the tire melts, but you will likely run into other troubles before this extreme is reached.