Can I move my washer and dryer within the laundry room?
Short answer:  Yes.  But it sounds like you do, otherwise you wouldn't be asking.

Long answer:  Check the manuals for your appliances.  The washer manual will tell you how far it can push drain water, follow it's instructions and limits.

The dryer will be more complex.  It will say it can push exhaust X distance, then give you math for bends, joints, vertical distances, etc.  For example, a 90 degree bend in a smooth-wall exhaust line usually counts for 5'.  So if the dryer can push 100', and there's one 90 degree bend, it can now push 95'.  Flexible pipe, like is typically used between the dryer and wall, is terrible for exhaust.  But again, the manual should tell you how to account for it.  Also take this opportunity to clean your exhaust, and plan to do so more frequently in the future.

Unfortunately, the correct answer to whether you can move these appliances depends entirely on factors of your home and the appliances themselves.  Where the exhaust lines go through the walls, pump or blower motor strength, etc.  Oh, and older appliances may not be able to reach full drain/exhaust distances.  Be careful.