Will Tyranitar be obsolete after Gen3?

No from memory he will remain the best dark attacker for a long time.

There are no exceptional Dark pokemon in Gen3. Highest level 40 max CP, assuming current formulas, is Crawdaunt at 2317.

In Gen4 there is Honchkrow at 2566 (Murkrow evolution), Weavile at 2815 (Sneasel evolution), and Darkrai at 3404 (legend). Also, depending on how they implement it in Go, Arceus can be changed to any type.

His first serious rival will probably be Darkrai in Gen 4, but he's a mythical pokemon that may have limited availability. In Gen 5 we then see Hydreigon, but based on stat totals he'll probably perform similarly.

Gen 6 brings mega-evolutions, Ash-Greninja, and Yveltal (which is a legendary), all of which will probably bring serious competition, but Tyranitar himself also mega-evolves.

In short, I think he'll always be useful.

I'm actually hoping that he gets better! He seriously needs a quick attack stronger than bite...

What makes you say that? He'll still be the highest CP dark and rock type in the game by a decent margin. He's the king of both his types and that's not changing soon.