Yes, muslin baby clothes can go in the tumble dryer, as long as you keep it on low heat. The loose, open weave that makes muslin breathable also lets air move through fast, so it dries quickly whether you tumble dry it or hang it to dry. Skip fabric softener and high heat. Softener dulls the breathability, and high heat can shrink the cotton over time. Ours ship pre-washed and soft, so you don't strictly need to wash new clothes before the first wear the way you might with other fabrics, though a first gentle wash never hurts anything.
Muslin creases easily straight out of the dryer, and that's normal, not a sign anything is wrong. It also softens with every wash and dry cycle, so a shirt that feels a little stiff when new feels noticeably softer after a few rounds through the wash. Ours are double-layer muslin cotton, sewn with flat seams that hold up well to regular washing and drying.
A gentle cold wash and a low tumble dry (or hang to dry, if you'd rather skip the dryer) is the routine we'd recommend for the life of the garment. Stick with that and the fabric keeps its shape and feel through normal use. See how long muslin baby clothes last with regular care for more on what to expect over time. Muslin holds up fine in a normal load with the rest of the baby laundry, no special cycle needed.