The changes would have to be accepted by at least miners, and maybe also by clients.
If the new code rejects something that was previously allowed and miners accept this change, then clients have no recourse. But if the new code allows something that was previously not allowed, then the network will split in two. All clients that refuse to upgrade will always follow their old rules, even if only 0.1% of the network uses these old rules.
So miners can't increase the money supply, change the difficulty calculation, steal money from accounts, etc. Miners could implement demurrage, increase fees, freeze accounts, etc.
It would be possible for clients to reject or discourage blocks that don't allow certain transactions, but this is very difficult to detect reliably, and it could end up fragmenting the network.
Lightweight clients like BitCoinJ can't do complete verification, so miners could trick these clients into accepting certain changes such as an increased money supply.
This is all very unlikely, though. The business model of miners depends on the existence and popularity of Bitcoin, so they won't accept stupid changes. Also, if the "official project" made insane core changes, it would lose its status as the official project. Bitcoin.org, weusecoins.com, bitcointalk.org, and the IRC channels are independent of the development project. All links can be changed within days.
2It is worth breaking this into multiple questions about the different types of possible changes. I could write six paragraphs on each of the following possibilities: Changes to the difficulty adjustment, changes to how many coins are created, changes that introduce transactions currently considered to invalidate a block, changes that introduce new transactions that users of standard transactions could ignore, and so on. (It's not entirely clear what "backwards-incompatible" means exactly. Do you mean the change would be mandatory to agree on a block chain?) – David Schwartz – 2011-09-22T22:25:17.153
Alright, as soon as I have time I'll break it up. – eMansipater – 2011-09-22T23:46:47.300