Yes, unless you've somehow borked your configuration (or compromised your private key), encryption will continue to take place reliably.
But you're ignoring one of the important (theoretical) goals of SSL - authentication, as in "So-and-so, who you trust, says I am who I say I am. So we're good to talk."
It's not a good idea to get your customers into the habit of clicking through frightening web browser warnings. If you're not going to fork over the cash to renew your certs, then consider securely setting up an in-house CA, and distributing your root cert to customers. (Well beyond the scope of this thread.)
On the other hand, if you have 'net facing websites with unknown numbers of customers - established and new - then you're ignoring expired certs at your own peril ($$).