Are there any audiobook apps that will remember where you last left off?
You can use Apple's iBooks app for iPhone/iPad to playback audio books and mp3 files and resume playback. With mp3s you may need to change the "media kind" option from Music to Audiobook in iTunes.

Import your audiobooks or mp3s to iTunes. Select your mp3 file in iTunes, right click and select Get Info from the pop up menu (or just press Command I) to get into the file's properties window. Click on the Options tab then change media kind from Music to Audiobook.

You can then sync your mp3/audiobooks from iTunes to your iPhone if you select to sync manually or sync everything