Hi cabert,
It would not provide good ventilation under the mattress so the chances of mold would be increased yes although by itself it may not be enough to present an issue depending on the other risk factors involved. There is more in post #10 here.
I would always use a mattress protector on a mattress. A mattress cover is part of the design of a mattress and is meant to keep all the parts of the mattress together and depending on the type can be a significant part of the feel and performance of the mattress. A mattress protector on the other hand is meant to protect the mattress from body fluids, accidents, and skin cells, and to keep your sleeping surface clean and hygienic. It’s much easier to remove and clean a mattress protector than a mattress cover. It also protect your warranty because even a small stain on a mattress (including a water stain) can void your warranty.
Which protector is best for you depends on which of the tradeoffs are more important to you.
A mattress pad is a different product once again and includes some kind of padding as part of its construction. While some of them will also protect the mattress they are designed to add some kind of cushioning to the mattress which can help to fine tune the performance of the mattress if you need it. A mattress protector is designed to have the least possible effect on the mattress while a mattress pad is designed to affect the feel and/or performance of the mattress.
Phoenix