What is the frame rate of the 1080p HDMI output on an iPad 3rd Generation?
For HD video, Apple's page on the specifications for iPad 3rd Generation states (emphasis mine):


  Video formats supported: H.264 video up to 1080p, 30 frames per second, High Profile level 4.1 with AAC-LC audio up to 160 Kbps, 48kHz, stereo audio in .m4v, .mp4, and .mov file formats;


So if it is capable of playing HD video up to 1080p at just 30fps, then all that an HDMI capture card with 60fps support can do is capture at 30fps (and maybe add interpolation frames to make it 60fps). Since there's no actual gain by doing this, a 1080p30 capture card should suffice.

On the apps side, according to benchmark tests by phonearena.com on the iPad 3rd Generation, it was able to go up to 60fps (depending on the test). But note that this is just graphics rendering frame rate, not video decoding or even video recording (the latter is just 30fps according to Apple's specifications page). Video encoding (converting the app's graphics to HD) cannot be better than the video decoding support of 30fps.

I do not know if the HDMI output would be at the same frame rate for apps and video.