Could you perhaps elaborate on your performance requirements?
I'm quite confident you will be able to get Tomcat running, here's a video of someone getting Tomcat 5.5 running on a beagleboard c4 (note the video is dated Feb 2011). Also remember that you can run a normal Ubuntu distro on the beaglebone, so installing Tomcat should be as easy as simply going 'sudo apt-get install tomcat7'.
The big issue is performance. If you're planning to use the beaglebone as a server handling thousands of requests a second you're probably going to have a problem.
The following table indicates the performance difference between embedded application processors and a desktop PC (the benchmark calculates pi to 500 digits, see Johannes Ernst's blog for more info).
The above benchmark obviously does not characterize other performance areas important in running a server (e.g. disk or network I/O), but let's assume you're looking at roughly a tenfold increase in processing time.
For instance, if a single Tomcat transaction usually took 10 ms to complete, it could now take 100 ms. If you're only using the beaglebone to serve a small number of consumers (infrequently), and responsiveness isn't as critical, it should work fine.