Recall that in order to be a valid block, it has to have a valid proof of work: the hash of the header has to be below the current target value (i.e. start with a sufficient number of zeros). If you just alter something in the block (e.g. the destination of the coinbase transaction), its hash will change and almost certainly will no longer meet the proof-of-work target. So you'll have to also find a nonce value which does result in a hash that meets the target. This is just as much work as mining a new block from scratch and carries no particular benefit compared to regular mining.
If you succeed in doing so, then there will simply be two candidate blocks for the head of the block chain: the original one, and your new "tampered" version (whose hash will necessarily be different from that of the original one). Miners will be free to choose either one to build their blocks on top of. Soon, a consensus will be reached as to which one is part of the block chain and which one is an orphan.