So, I was just wondering whether the spoilers on a 737-800, after automatic deployment when landing, will go back down automatically, or need manual input by pilots for the spoilers to go down.
The spoilers on the 737 after auto deploy will re-stow automatically based on TLP (thrust lever position). This is a function for a rejected landing which will automatically lower the boards & retract the levers for go around... most pilots after taxi off the runway quickly advance the throttle to let the actuator reset the system in leu of pushing the lever forward (it has a clutch in the linkage to allow manual positioning as well)
It should depend on the aircraft. In the EA-6B Prowler the spoilers deployed when 3 conditions were met: Switch armed, weight on wheels, and throttles at the idle stop. (this presumes hydraulic pressure is available to actuate them...)
Adding any amount of power, even just enough to taxi, would cause them to retract if they were armed. The post landing checklist called for the switch to be turned off, so they would then retract and not fluctuate up and down during taxi due to throttle movement.
I would imagine others work similar to this, but if you want a more detailed answer for a certain aircraft you will have to specify which particular one you are interested in.
To add to Michael's answer, on pretty much all modern airliners they retract automatically after completion of the landing and are disarmed until re-armed on a subsequent take-off. There will also be some kind of manual override to manually disarm them.
There is usually a timer cycle of say 30 or 40 seconds after deployment, while still on the ground, at which point they retract and disarm. It's the end of this timer cycle you are seeing when the spoilers come down as you are pulling onto the taxiway.