Recently I tried to run Tomb Raider 3 in a higher Resolution than 1080p. 1080p worked fine.
The highest I got was 2048x1536. If I choose a higher value in setup, the game crashes.
Any ideas?

According to wsgf, you have to se dgVoodoo 2 to remove the resolution limit.

Somehow I got it to work now, except that all my textures are gone.

Can you post a screenshot of the game setup? Particularly is "8 Bit Textures" enabled or disabled?

I suggest you to take a look at this thread.