I am having a problem with my guitar. I have a bit of fret buzz that is just unacceptable. If I play the open low E, there is no buzz. I can fret the low E from the 1st fret to the 5th and get no buzz, but past the 6th fret it starts to buzz.

There is a dip in the neck, meaning that the middle of the fretboard is lower than highest and lowest fret.

Do I tighten or loosen the truss rod to fix this problem?
Tighten the truss, but if you can stand it try raising the action first (by a little) to see if that remedies the problem.
My Gear:

Ibanez RG 7 EXFX (27'' scale, 2x EMG 707)

Mesa/Boogie Studio .20 Caliber

Line 6 Studio POD GX
What the previous guy said - tighten the truss rod & raise the action by raising the bridge.
Rock [James] Roll
tightening the truss rod or raising the action might help, or the frets might not be level in which case getting the frets levelled by a tech would be the go.