If the landlord hires the pest control company to exterminate bugs and mice then yes they will be responsible for covering the cost of the exterminator to do their job.
If you "the tenant" hire the pest control company without first contacting the landlord about the pest problem then you the tenant will be liable and responsible for the pests control bill.
If you made contact with the landlord about getting rid of the bugs or mice or other pests and they do nothing about it and then you have to hire a pest control company then the landlord just might be responsible for paying the pest control bill.
You'd have to take the landlord to court over that though and you'd likely eventually be evicted after taking your landlord to court over something like that.