Should I stop applying faculty positions after I take on a postdoc job I may have a postdoc job offer. The professor for the postdoc still hesitates to make a final decision and asked me if I can confirm my commitment (working for at least one year). I have finished one-year postdoc (not at the professor's institution) and have been very actively looking for faculty positions.

If I make the commitment, does that mean I have to stop applying faculty positions, at least for the first year? I am not sure if this is common practice for postdoc-supervisors.
Don't stop exploring other positions until you have a firm commitment from the professor. The situation is mutual. You both have to agree before anything is finalized. Until then you are still free to consider all options. 

By firm commitment, I mean something that you can put your signature to.