There is some specialized equipment for television repair, and much of it is getting harder to find as CRTs become less common (prices will go up).
For CRTs, here are a few of the more common specialized test gear you might need:
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A CRT tube tester/rejuvenation machine (Sencore CR7000 is a popular choice)
A CRT test pattern Generator (such as a Sencore CG169, though any such generator should be fine)
High Voltage Probe (CRTs usually run upwards of 25kv)
Blue Ring Tester (for testing transformers, such as flybacks)
Some more general, but necessary, pieces of gear would be:
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Oscilloscope w/10Mohm 10x Probe (an old scope is fine, but bang for buck the best buy right now is the rigol ds1054z, hands down)
Function Generator
Frequency Counter (most modern scopes have this built in)
Several good Digital Multimeters (need at least two, to measure voltage and current at the same time)
ESR Meter (to test caps, which are a common failure in pretty much all TVs, old and new)
There are a lot of additional things you need, but those seem obvious enough that I won't list them (soldering iron, lighted magnifying lens, screwdrivers, etc) but honestly, one of the most important pieces of gear you will need is the service manual for the chassis you are working on. You can get away without some of the gear above if you have patience and a schematic, though the above can make your life much easier.