Individual miners have different standards for what fees they will accept. Most simply aren't going to process a transaction without fees, so you'll be waiting for somebody to generously record your transaction to the blockchain without compensation. I've never tried this, but I've seen people say that they've had to wait days. It will vary, of course. If you get impatient you could always propagate another transaction that spends the same exact funds, but allowing some of them to be used as a fee.
I've heard estimates of how many transactions the blockchain can handle that range from 3 to 7 per second. I'm really not sure what the average transaction size is, but they all have to fit in a 1 megabyte block. Even a miner who generously records transactions without a fee is almost certainly going to give priority to transactions with a fee attached, so some of your transactions might get recorded and the rest left to wait for another generous miner to get a block. Unless the inputs of one transaction require the outputs of another, you won't even be able to know which ones will get recorded first.
1Thanks for the answer and references.
My question was mostly about the impact on Bitcoin itself, and (un)fortunately it seems like these current spam attacks show that no, Bitcoin does not handle such large quantities of transactions well. According to reddit, there are 11k unconfirmed txs with a 21MB mempool right now, and a ~0.0004 BTC fee is needed to get into the next block.
If I were to sent thousands of txs with no fee every hour, they'd take forever to go through. – Soricidae – 2015-07-29T06:04:23.957