If you remove a tick but leave the head in skin will this cause a problem?

No: If part of the head or any mouth parts of the tick remain in the skin after removal of the body, these should be left alone as they will be expelled on their own. If you try to remove these parts, significant skin trauma may result.

See below: Usually not, but the site can become secondarily bacterially infected and if the tick had tick-borne illness, this could still be transmitted. Basic rule: watch the site and yourself for changes.