We consider a singly quantized off-axis straight vortex in a rotating dipolar ultracold gas in the Thomas-Fermi (TF) regime. We derive analytic results for small displacements and perform numerical calculations for large displacements within the TF regime. We prove that the dipolar interaction energy increases (decreases) as the vortex moves from the trap centre to the edge in an oblate (a prolate) trap. We show that the angular velocity representing the onset of metastability is lowered (raised) in an oblate (a prolate) trap. We find that the effect of the dipole-dipole interaction is the lowered (raised) precession velocity of an off-centre straight vortex line in an oblate (a prolate) trap.