In most situations a police officer needs a search warrant to enter your home without your permission.
Although if they're chasing someone down who is a danger to the community and the person enters your home and hides out then the police can then enter your home without a warrant to apprehend the person and take them into custody.
Also if you continue to hide the person out you can be charged with aiding and abetting and could wind up in prison yourself.
If someone was on the run and has done something illegal and bad and that person hid in my house the police are more than welcome to go into my house even if I'm gone and even break down doors or whatever if they need to get the person they're looking for.
If it's not an emergency or something is in plain view then the police don't need the warrant to enter your house.