Under what conditions do I need to have a takeoff alternate airport listed on a dispatch release? How far away can my takeoff alternate airport be? Do I need to worry about this in part 91 operations or is it just a 121/135 thing?
You should refer to the published IAPs for an airport to determine if it has non-standard takeoff minimums (for AeroNav charts, this is the T in the black triangle). There's a section on this in the Instrument Procedures Handbook, Chapter 2 (2-6):
The FAA establishes takeoff minimums for every airport that has published Standard Instrument Approaches. These minimums are used by commercially operated aircraft, namely Part 121 and 135 operators.
No person may takeoff an aircraft under IFR from an airport where weather conditions are at or above takeoff minimums but are below authorized IFR landing minimums unless there is an alternate airport within 1 hour's flying time (at normal cruising speed, in still air) of the airport of departure.
...and so does 14 CFR 121.617:
(1) Aircraft having two engines. Not more than one hour from the departure airport at normal cruising speed in still air with one engine inoperative.
(2) Aircraft having three or more engines. Not more than two hours from the departure airport at normal cruising speed in still air with one engine inoperative.
14 CFR 91 does not require a takeoff alternate. I'm going to quote the IFH again because it's an authoritative FAA source and because I very much agree with it (2-6):
Aircraft operating under Part 91 are not required to comply with established takeoff minimums. Legally, a zero/zero departure may be made, but it is never advisable.