So, I cannot seem to open a chest in Minecraft Pocket Edition. I tried clicking on it, which resulted in the chest breaking or not doing anything. I haven't run into that problem on the PC.
If you are in creative, as of the latest version (Alpha 0.8.1) when this question was posted, you couldn't open chests in creative.
Some other people might but I think you can't because there is an update that you need to do to be able to get hoppers and redstone stuff and also letting you get into chests and other things, too! (I don't know if you can, but I think you can do it in survival)