While most of the 3D printers on today's market are almost maintenance free, there will come a time when you will need to tighten a belt, change out a broken / worn out belt, or even perform surgery on the extrusion head if a heater goes out or a catastrophic filament jam occurs.
In the 75-pounds of plastic I have printed in the last two years, I have only had to work on my printer a total three times, one of which involved disassembly of the extruder to clear a bad filament jam. So, while rare, things do happen that require some tools that you may not have just laying around.
Most 3D printers use some form of cap screw to hold things together, and often times, those cap screws utilize Allen heads, also known as Hex screws. The Allen head screw is a very good choice, as it allows great torque to be applied from the wrench, with little risk of stripping or slipping, and is one of the main reasons builders chose them. A very good set can be picked up from a hardware store or harbor freight for under ten bucks, though I recommend getting a set that utilizes the ball-head design for added flexibility.
I think everyone knows what a screwdriver is, and more than likely has a few lying around their home, but not everyone has a decent micro-screwdriver set lying around. In the world of 3D printing, the fasteners can come in sizes ranging from large to very small, and a good set of micro-screwdrivers is an excellent thing to have.
One specific place these will come in handy is if you ever need to adjust the small trim pots on the controller board that control the stepper motor current, which often require a very small flat head screw driver. A very nice set of these usually runs a few bucks, and will last a life time if properly cared for.
Small Wire Bristle Acid Brush
This is one of those tools you would never know that you needed until you needed it. Sometimes filament gets tangled on the spool, or something will happen that causes the feed gear or hobbed bolt to chew into the filament, which will clogs the teeth.
Without one of these simple brushes, cleaning the teeth is an aggravating, and tedious task. With the brush, the chore becomes as simple as activating the extruder motor, and using the brush to wipe away the filament chunks.
Nozzle Cleaning Needle
While not a necessity, a needle the same size as your nozzle's diameter will come in very handy in the event of a jam that involves charred plastic, or something foreign making its way into the heat chamber and burning. These can be picked up from places like Amazon, McMaster Carr, or any other source that sells industrial testing equipment. Personally, I have never had to use the one I own, but I feel better knowing that I have it on hand in the event that I might need it.
If you built your on 3D printer using threaded rod, or your extruder features a removable nozzle, a wrench set makes life very easy. If you plan on using a wrench to change nozzles, another wrench sized to fit the heater block will make changing nozzles easy. While you could take the easy road and purchase two crescent wrenches, I highly recommend buying standard box end wrenches of the proper sizes, as this ensures the best grip possible on the parts.
IR, Laser or Thermocouple-based Thermometer
When trouble shooting issues with your printer it is always best to know if your nozzle and heated bed are making it up to the desired temperatures, and this is where a thermometer comes in handy. Personally, I prefer a Type-K Thermocouple connected to a multimeter, but a non-contact IR or Laser-based thermometer will suffice.
One important thing to remember is that IR and Laser units are not 100-percent accurate when it comes to shiny reflective metal surfaces, so keep that in mind. A Type K thermocouple can be gently taped to the nozzle or heated bed using Kapton tape, for a very accurate temperature measurement.
When troubleshooting electrical issues on your 3D printer, your best friend will quickly become a good multimeter. You can use it to test everything from continuity between connectors, heater resistor issues, thermisitor accuracy, and even to see if you have a bad stepper motor or driver.
A decent multimeter can be had for about $30 at Harbor Freight, or from a hardware store for about $50. To get a top quality meter, expect to pay between $200 and $2,000 depending on the features and quality you would like to have.