Is this some sort of disease? Some of my fig trees have some part which is dried, looks like a dried wood but rest of the branch is green. Should I cut those branches or leave it like that?

Thanks a lot!

Most likely some sort of mechanical injury either from rubbing or deer, but really can't say for sure.  If you cut it wait until the spring.  Don't want new growth to start below the cut this time of year.  If you don't cut, it may be fine.  It only needs a portion of the bark/cambium layer in order to effectively relay nutrients up/down.  But if it bothers you certainly you can cut it and you will get new growth sprouting from the nodes below.

Thanks Rob!