I think the conventional wisdom is that yes, it will likely increase your interior case temp a bit. However, if your case has good "fannage" and an effective airflow scheme, adding some good quality dampening mats isn't likely to raise the temps so much that the bad would outweigh the good. Obviously, the more dampening material you use, the more your case will be 'insulated' and thus retain heat. How much quieting you achieve depends on the quality of the dampening material and how much you use (of course).
Below is some info I copied from SPCR's Web site. It's from a very
loooooong article on quieting down PCs. At the end I've included a link if you wanna see the original article, which has hyperlinks that didn't transfer over in the verbiage below. Maybe you'll find it useful. Good luck.
9. Acoustic Dampening material
Acoustipack is a heavy sound deadening material you can apply to your case. There are several versions, including pre-cut kits for some Antec cases and ones made specifically for HTPC cases. If you have already have a quiet system, it won't that help much. If you have a loud system, especially with high frequency noise or a lot of case vibration/resonance, it should be a significant help. The Acoustipack Deluxe V.2 comes with extra foam blocks and a better sheet of acoustic dampening material than the normal version, or versions that are pre-cut for particular case models. A warning, most dampening materials will add significant weight to a case, so expect your computer weight to increase by 7-10lb.
Mutemat seems to be similar to Acoustipack according to the SPCR review, but there is no definitive answer as to whether it is better or worse than the previously reviewed Acoustipack. The reviewer didn't do any "scientific" testing, but he believes that it's about the same quality as Acoustipack. As reported in review, this is made by a U.K. company, and should be easily available to people there.
Nexus Damptek has gotten a couple of feedbacks in the forums. The Nexus kit has the blocks for 3.5/5.25 bays sold separately from the sheets that are cut and mounted to the walls of the case. The blocks are reportedly easy to tear, according to nicci, so you should avoid purchasing them. I?ve read somewhere that the foam is somewhat meh in terms of effectiveness and somewhat hard to remove after installation, unlike Acoustipack.
For those in Germany, Silentmaxx also makes a dampening kit. No users on this site have reported using it, so it's unknown how good it is. See here under Produkte
The four above are expensive in the $60-80 range. Cheaper stuff like Paxmate (around the $20 range) is useless. Buy the good stuff, or not at all. Unfortunately, the four dampening agents above have not been tested in head to head comparisons, so it is a mystery as to which may be the best. Alternatively you can try to make a DIY dampening kit from Mcmaster-Carr as outlined here using Sorbothane, a very popular material used to decouple hard drives and dampen cases on these forums, and a cheap foam such as egg-crate foam. Others have used floor tiles from Home Depot and the like.
Link to original article -- scroll down a long way for the section on dampening materials