Will Bitcoin suffer from a mining Tragedy of the Commons when mining fees drop to zero?
There will definitely be a tragedy of the commons problem if things stand as they are now. This was discussed at some length here and elsewhere.

There are some proposed ways to address this and make transaction fees nonzero (block size limit, hardcoded fees, insurance entities, mining cartel, gentleman's agreements which are maintained for fear defection would beget more defection, etc.), but I don't think either results in the combination of competitive transaction fees and a level of hashrate which is enough to offer reasonable security.

This means that there must be some way to improve the network's security at a given hashrate. The only way I know to do that is by augmenting proof-of-work with proof-of-stake, discussed for example here. This is based on the observation that obtaining a majority of the bitcoins in existence is orders of magnitude more difficult than obtaining a majority of the hashrate.