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Thanks everybody for your feedback.
I had previously only tested controlling applications other than EyeTV with the EyeTV remote control, which worked fine.
Thanks to your feedback, I could reproduce and eventually fix the issue. A new version of Remote Buddy is now online as a silent update. I've appended the release notes with this info:
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Fix: Workaround for what looks like a bug introduced by Apple with OS X 10.6.5 (rdar://8672139) that breaks communication between EyeTV and Remote Buddy regarding EyeTV remotes. Note: This fix was not part of the original 1.16.6 release, but was released as a silent update (on 16th November 2010 at 10 am UTC+1). If you are affected by the problem, visit http://www.iospirit.com/products/remotebuddy/download/ and follow the instructions to re-download Remote Buddy and replace your existing copy with the updated version.
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Best regards,
Felix Schwarz