Do you have to accept a postdoctoral position if the university has paid for accommodation and international flights for you to come and present? I have been invited to give a seminar related to applying for a postdoctoral position. The professor has been very kind and seems  like a very nice person. He also helped me look for a postdoctoral position for my girlfriend. Now I have received the tickets to travel to the university with first class accommodation and luxury amenities. The trip involves  international travel and also that I have been talking with other professors to see my best options.

Does this force me to accept the position or is it common that researchers give these nice invitations?
Do I need to mention  to the professor that I am exploring other options?
If you are getting excellent treatment, that likely means they are fairly keen to hire you.  They are working to recruit you.  Accepting nice accommodations and amenities from them during the hiring process places no obligation on you--none at all.  That being said, this sounds like it might be a really good place for you.  Use your judgement about what will be optimal for yourself.