i agree with all the above, the main thing to protect your bird here would be to use tube feeders and drinkers,
this will stop the mice having acess to the food and water an durinating into it..
mouse urine is poisonous to birds so if you suspect a mouse has been in the seed then throw it away and get fresh, put in sealed plastic containers with an airtight lid...
if an heavy infestation is their then as said use sticky traps around the edges of your walls, this is a mouse's favourite place to run and shelter from the human eye... find where the mouse goes.. this isnt difficult with heavy infestation (follow the poop) if their are any mouse holes coming in from outside use quarter gage mesh as mice can not get through this at all ..
but i would also call the exterminator out to sort the infestation, tell him you have birds and he will then use the safest method to prevent the bird getting at the poison (usually warfarin) in the form of coated canary seed!! where birds are not present that is..
their are bait blocks you can buy or the exterminator can use them aswell as sticky traps ..
the bird will not be the cause of the mice, the mice will go whereever their is food... the mice can even live in your house because it is close to the next doors house which contains all the food they want.. all because you have them doesnt mean the mice are their for your food scources...
some mice live outside but come in for warmth or food so may just be visitors but once it gets cold more will accumulate and they will breed....
i am guessing you have seen these mice? if so what color are they as they could be a house mouse or like in my house they could also be field mice as i had a mouse problem once.. my 4 year old son caught one and wanted to keep it as a pet
turned out the mice had been desturbed when the neighbours had a new kitchen fitted and they ripped the ceiling out so they decided to take residence at mine!!
however you dont say what sort of bird you have .. if you told us that that would help as while the infestation is been cleared you may (if you have small birds ie finch or canary) have to ask somebody to look after the lil feller for a week as once any food scource is cleared up the mice will be hungry and angry.. small birds can and do fall vicitm to mice attacks,
i saw a mouse in my birdshed and the mouse was literally dragging the poor little finches about and shaking them.. (alas the finch died)
however for normal size birds ie budgies , cockatiels, and bigger they will be fine and no need to move them..
good luck and hope you sort your problem soon
katie