Some remotes are actually slow! I got a TiVo from ebay that came with a worn orginal remote and a new orange replacement remote. The TiVo was very slow scrolling up and down the menus, so I tried two orignal remotes (including the worn one that came with the TiVo) and it was very snappy, going up and down menus as fast as on my original TiVo. I used the same new set of batteries in both remotes. The electronics inside the orange remote were slightly different, as you may expect from a copy/clone remote.

It's got nothing to do with the problems described below, but I thought I'd mention it, just out of interest.

NoSignal it is normal for the parts you describe to get quite hot, I doubt that is the reason your TiVo keeps locking up. Drop me a line, and I'll give you a hand with your TiVo.