Consider calling the person who takes a refugee into their home a host. The place is simply termed a place of asylum, where asylum is used in the sense of the definition below. (The country can also be called a host.)
host noun
1 A person who receives or entertains other people as guests.
- ODO
asylum noun
1 (also political asylum) mass noun The protection granted by a state to someone who has left their home country as a political refugee.
‘she applied for asylum and was granted refugee status’
- ODO
Here's a wikipedia entry that uses the term country of asylum in this manner:
Here are a couple of examples where the word host is used in this context:
Becoming a host is an opportunity for you to show solidarity to people in need and help in a very real and practical way.
- Refugees Welcome
But there's also a lot of relief and pleasure for the host and guest. Living with refugees neutralises fear and loathing.
- The Sydney Morning Herald
Here are a couple of examples where the word host is used as a verb. This is arguably a case of zero derivation, and we can call the person or place doing the
hosting a host.
Due to the incredible surge of interest in people offering their homes to host a refugee, Alyan’s List organisation has been developed in partnership with Australian Homestay Network to implement a placement program.
- Australian Refugee (note: host here is used as a verb)
Ten countries host half of world's refugees
- News Grid