I'm not sure what's going on here, but I've generated a test case and will upload it shortly.
Created attachment 633132 [details] Fixed test case
Steps to reproduce: 1. In Fennec Native, visit https://bug764817.bugzilla.mozilla.org/attachment.cgi?id=633132. 2. Type 'h' into the text field. 3. Tap the 'add letter a' button. The expected new value of the text field is 'ha', while the actual value is 'a'. You can also try this out in Mobile Safari, Mobile Chrome Beta, and the stock Android browser; all of them exhibit the expected behavior.
I think this is ime related. I can reproduce this on the Galaxy Nexus, it underlines the 'h', which means some ime thing is going on. I can't reproduce it on the Galaxy SII, which doesn't do that ime thing.
This works for me using Firefox 37.