Michagogo worded it very well here:
http://www.reddit.com/r/Bitcoin/comments/22i9t1/psa_regarding_the_heartbleed_bug_cve20140160_and/There are exactly two places in Bitcoin Core that may be affected by this issue.
One is RPC SSL. If you're using this, turn it off. If you don't know what that is, you most likely aren't using it.
The other is the payment protocol. Specifically, fetching payment requests. If you're using a vulnerable version, do not click any bitcoin: links and you will be protected. Note that this is only relevant for the GUI, and only for version 0.9.0.
If you're using self-built executables, you're most likely using dynamically linked OpenSSL. Simply upgrade your OpenSSL package and you should be fine. If I'm not mistaken, the same applies if you're using the PPA. If you're using release binaries, a version 0.9.1 is being prepared that will use the fixed OpenSSL 1.0.1g.
Note that if you're running the GUI (p.k.a. Bitcoin-Qt) you can check your OpenSSL version in the debug window's information tab. If you're on anything earlier than 1.0.1, for example 0.9.8, you're safe. If you're on 1.0.1g or later, you're safe. If you're on 1.0.1-1.0.1e, you may be vulnerable. However, that may not necessarily be the case -- for example, Debian has released an update for Wheezy, version 1.0.1e-2+deb7u5, which fixes the security bug without bumping the version number as reported by OpenSSL.