If my boyfriend came here on a tourist visa but we are planning to get married, but his visa expired, are we able to marry without any issues?

The answer will depend on a lot of things, such as whether his entries were legal, and whether he disclosed the true reason for his entry.

Generally, a foreign national who enters the U.S. lawfully and with inspection, such as with a visitor's visa, and who then becomes married to a U.S. citizen, may succeed in a marriage-based adjustment of status case to become a Lawful Permanent Resident (to get a "Green Card"). This is true regardless of whether the initial visa may have expired, and regardless of whether the marriage took place before or after the visa expired. Of course there are many additional details that determine eligibility. It would be wise for you and your fiance to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.

Usually yes, you should be able to sponsor him if he entered the US legally, but you should consult with an attorney.