Can you swim with a titanium necklace on?

It depends on the quality of the titanium. Low-quality titanium alloy might corrode in chlorinated pool water or sea water. For the most part, high-grade titanium won't tarnish even when exposed to salt water and pool chemicals for long periods.

If the necklace has other elements such as chains attached with other types of metal. Consider how those other metals reacts with pool and sea water before jumping in with your necklace just to be safe.