No, you will not be convicted of obstruction for refusing to grant the police permission to enter your home when they do not have a warrant. Now, they can arrest you, wrongly, illegally - as sometimes happens unfortunately. BUT, you do not commit a crime by exercising a constitutional right to refuse them entry into your home.
Now, if they waited and then got a warrant and found him inside, things have changed. Now you have done more than just exercise a constitutional right. Now you have done that and at the same time you have harbored a fugitive. That would be obstruction. So, refuse them entry without a warrant, but don't have him inside when they get in.
I hope that clears things up a little.
Possibly arrested, but very unlikely that you would be convicted.
The information provided in this answer does not create an attorney-client relationship and is not considered to be legal advice.
Based on the facts you present you would not be arrested for obstruction. It is your Constitutional Right to keep the police out of your house without a warrant barring certain exceptions which don't seem to apply here.
James L. Yeargan, Jr. is licensed to practice law in the State of Georgia. All information given is based only on Georgia law, and is not directly applicable to any other jurisdictions, states, or districts. Any answer given assumes the person who asked the question holds a Georgia Drivers License, and this license is not a commercial drivers license (CDL). This response, or any response, is not legal advice. This response, or any response, does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information. Any state specific concerns should be directed to an attorney who is licensed to practice law in that respective state.
You can be arrested for whatever is on the officer's mind- unfortunately. Whether or not you'll be convicted is a whole other matter, and as many other attorneys have pointed out, that will probably not be the case. If you have been arrested, I highly encourage you to get an attorney to protect you from further injustice.
No legal advice is given here. My responses to questions on Avvo are never intended as legal advice and must NOT be relied upon as if they were legal advice. I give legal advice ONLY in the course of a formal attorney-client relationship. Exchange of information through Avvo's Questions & Answers forum does not establish an attorney-client relationship with me. That relationship is established only by joint execution of a written agreement for legal services. I am only licensed in the States of Georgia.