Does the Shimano Tourney TX rear derailleur support an 8 speed cassette?

Any Shimano 7-9 speed rear derailleur will work with any 7-9 speed Shimano shifter. To upgrade to 8 speed, you need an 8 speed Shimano-compatible cassette or freewheel which works with your rear wheel, 8 speed shimano shifters, a 7-9 speed Shimano rear derailleur and a Shimano road front derailleur if you have road shifters or a Shimano mountain front derailleur if you have mountain shifters.

The rear derailleur needs to have appropriate capacity and sprocket size tolerances for the gearing you want to run, and the front derailleur has to be matched to the number of chainrings, size of the jumps between chainrings and chainring sizes. Both have to have apppropriate mounting as well, obviously (tourney comes with a claw adapter in some cases; most modern frames don't need it).

So, yes, you can use a Tourney rear derailleur with a Shimano 8 speed shifter, but you have to make sure the rest of the components work. That being said, upgrading from 6 or 7 speeds to 8 speeds usually doesn't improve anything (esp. with low quality parts like Tourney).

I am currently using a Shimano Tourney TX RD on my old mountain bike that has 8 speed casette, no problem in either shifting response and performance.

I have a Shimano Tourney 6/7 speed that works on a 8 speed cassette with Altus shifter. This comes stock with the bike. If you search online, there are folks that tried with a 9 speed cassette and it worked for them. As always, YMMV with the shifter you use, and I suggest you check your combination online first before taking the plunge. Or if you have spares in your parts bin, you could always strap them on and try.