Can I safely make a door frame taller?
Remove trim.  
Remove door casing.
Remove drywall around jack studs (2x4 closest to door opening on each side) and pretty much all the way up to the ceiling.  Stop if you have roofing structure above your door - keep going if you don't.    
Pull out jack studs (you could just extend them with a piece of 2x4 on non load bearing walls)
Push up header to desired height.  You will have to take out the cripples (small 2x4s) above the header.  I don't think the door in your picture is going up a foot with that trim.
Put in new longer jack studs on each side.
Install new, longer door.
Drywall, mud, tape, all that good stuff - this might take the longest.
Install new, longer trim.  You could use old trim if you could find some casing that will fit on the bottom corners and if door is same width.