I know this is an old post, but I'm currently having the issue as well. I first noticed it about 6 months ago.
I work on a Mac where everything looks fine, and the problem seems to only occur on Windows (I've seen the same issue in IE 10). And, my guess is that it has to do with transparency.
The "bleedthrough" seems to just occur with anti-aliased pixels. The white lines that appear are where the bounds of the text box are located — for me anyway.
The confusing part is that even if you flatten everything the white lines still appear which makes me think that Photshop is hanging on to something in the history that gets saved out. But even if you clear the history the lines are still there.
The only "fix" I have found is to save each frame as a jpg and then create a new animation out of those frames which is a pain and a hack.
What a mess.
You are doing nothing wrong and it is not a Photoshop issue, but a Microsoft one from what I understand after researching this earlier this year. If you were to check you would NOT have this problem in Firefox on a Windows computer, only IE. I believe it started with version 8, but can't swear to that, but I know for sure it was in IE 9. At any rate the way to "fix" it is to change the Compatibility View Mode in IE. Depending on how this view is set you may see any number of issues including the lines that you descrbe. I've included a screen shot showing where to change this. (The small icon next to the address bar.) On my computer when the icon is blue it displays incorrectly, when it is gray things look as they should. Or you can use Firefox and avoid the problem all together. We even had issues where it made our web content reflow around the graphics incorrectly if we didn't have this set correctly. What's bad about it is that it's hard to explain to your customer why their stuff looks wrong and Microsoft has known about this issue for quite a while and hasn't fixed it.
Glad you posted on here so I could post back a fix. I usually do when I find a solution, but it took us several months to come up with what was happening and I'd forgotten to post back.
Good morning! I know this post is pretty old at this point, but recently I was having the issue with hairlines in my animated gifs. Nothing, and I mean NOTHING fixed this (including changing the compatibility view unfortnuately ) until by chance I noticed that my preview in internet explorer was set to 105% zoom. Once switched to 100% the hairlines were gone! A coworker of mine was also seeing the hairlines and we discovered his preview opened at 95%, and again once switched to 100% all was well. This may, or may not, help you (hopefully it does) but I thought I'd take the time to let you know what I saw on my end. Animated gifs appear to have image bleedthrough in Internet Explorer 9