Permitting weather, most airline pilots get visual landing clearance in the US, unless they specifically have requested an instrument or RNAV approach due to their company rules or special circumstances. My question is: when they disengage Auto Pilot for visual landing, do they still use Flight Director until short final?
This answer will be quite anecdotal (I have not seen a survey on the question). It will answer the question for "most people I've worked with", and for "my" aircraft (B737NG). Also, the usual disclaimer: different airlines/aircraft manufacturers may have different SOPs/instructions/limitations. Consult your manuals.
In general: most pilots will leave FD on, IF it still provides useful and reliable information, without being an obstruction/nuisance (i.e. when configuring it to give correct guidance would be more trouble than it's worth, given that you already have visual references). This philosophy is also usually what you'll find in an airline SOP, but in more formal language.
I'll try to provide some examples (non-exhaustive) to illustrate this depending on how you were set up before being cleared for the visual approach:
ILS to straight in landing
Leave the FD on until landing (on my aircraft, B737NG, it will disappear at 50 ft RA), both localizer and glideslope will give usable and reliable information to the FD. (Caveat: in visual conditions, ATC might be less strict about avoiding signal interference. They might for instance let an aircraft on the ground taxi to cross your runway while you're on approach, which could cause localizer/glideslope interference).
ILS to sidestep
Most would turn the FD's off. Both localizer guidance and glideslope guidance will be to the "wrong" runway. The FD is likely to be more of a nuisance than a useful tool.
Non-ILS approaches with no centerline offset
The Localizer/LNAV path will give good lateral guidance to the FD. The vertical guidance will depend on the approach and your equipment. If the aircraft generates a good quality vertical path, which crosses the runway threshold at a reasonable height (50 ft), there is no real good reason to turn the FD's off. If all you have is vertical speed (V/S) guidance, it might be more helpful to turn FD's off.
Non-ILS approaches with centerline offset (for example most VOR approaches)
At some point during the approach, you'll want to abandon the approach course, and line up with the runway. Where this point occurs depends on specific approach, and how you intend to fly it.
Example: A VOR approach to runway 36 (runway heading 000) might use a VOR 1 nm east of the runway with inbound course 005. You might want to stay on the inbound course of the VOR until it crosses the extended runway centerline. In this case, most people would keep the FD on until reaching this point, and then turn it off, as it won't give you any useful guidance afterwards.