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At my last hive inspection I noticed what I thought was queen cups and possibly swarm cells. Should they be removed during inspections or not? Also, if they are not removed what are the implications? Thanks.

Your second question is irrelevant. If you do remove them, your hive will likely end up queenless and dead. That implication is the one that matters most.

If your bees are building swarm cells (queen cells near the bottom of brood containing larvae and royal jelly) and not just empty cups, they may have made the decision to swarm. The swarm cells provide the replacement queen after your original queen has left. Destroying the cells won't necessarily stop the bees from swarming, and if they leave and you have destroyed the swarm cells, you may have doomed the hive to queenlessness.

More wordy, but essentially what Wally was saying.

And if, by any chance, they are supercedure cells, same story. You have killed their replacement queen and the hive dies.

Same result either way.