What happens on the blockchain if I try to transfer bitcoins without providing a recipient address?
Most wallet software won't let you send to a null address. If you have a transaction without an output script, that's invalid and won't be accepted by the network. If you have an output script that's simply an OP_RETURN or otherwise unspendable, those coins are lost forever.
Generally, you should be using a wallet to transfer bitcoins and if you don't provide an address using a wallet, you'll most likely not be able to send it.
To move a Bitcoin balance, you have to create a valid transaction. Bitcoin transactions have at least one input and one output.
Most wallets only allow creation of Pay-to-Private-Key-Hash (P2PKH) transactions. The output of P2PKH transactions must specify a recipient address. Leftover input balance that is not assigned to an output will be claimed by the miner as transaction fee. Therefore, a P2PKH transaction not providing a recipient address is invalid.
To create another type of transaction, you must either create raw transactions or use other advanced features. Using advanced features allows for more exciting ways to shoot yourself in the foot. Alternatively, you could send money to an address whose private key is unknown, or which can never be spent. This is used e.g. for Proof of Burn.
TL;DR: in most cases you shouldn't encounter the problem of "transferring bitcoins to no address".