Are there any specific type of FDM 3D printers that I should look for?
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No. Not all 3D printers can print flexible material.
The first place you will have trouble is in the extruder itself. Flexible filament will find any way to escape from the confines of the rollers and the guide tube. Any opening will allow the filament to buckle and find a new path. It is common to find that the filament has filled any gaps inside the extruder, and wrapped itself tightly around the drive roller.
@0scar describes this in his answer. If the extruder does not have tight tolerances, with 1/4mm or less clearance between the drive roller on all sides, AND if there isn't a tube mated right up into the place where the filament is pressed between the drive roller and the idler, your extruder will fail to print Ninjaflex (a very flexible filament).
Oscar also mentions that Bowden feed doesn't work well. It is fundamentally the same buckling problem. There is always a gap around the filament inside the Bowden tube. A still filament will rub at some points, but a soft filament will compress and ripple, filling the entire lumen inside the tube, and greatly increasing the friction. Higher friction means more extruder pressure, so more buckling, more friction, and more failure.
Even if you are lucky and don't experience the run-away friction problem, the flexible filament is more compressible. With a Bowden feed system, that compression must be preloaded at the beginning of each extrusion movement and relieved for each retraction. With direct drive, the filament also compresses, but the amount of filament is much less, so the compression is less, and the problems are less.
Not all printers are suitable to print flexible filament. E.g. 1.75 mm filament printers with a Bowden extruder/hotend combination will not work (you may have more luck using 2.85 mm filament, which is stiffer because of the increased diameter). For 1.75 mm filament you require a direct drive extruder, e.g. with the stepper mounted onto the hotend, even then some additional guide parts need to be printed to make it work. This also depends on the amount of flexibility of the filament, some are more flexible than others.
E.g. Ultimaker 3D printers use 2.85 mm filament with a Bowden setup. They also sell a flexible filament that can be printed with these printers. Even for direct drive extruder printers like the Anet A8 (a cheap Chinese Prusa i3 clone) inserts exist (e.g. this or this one) to even better guide the filament to prevent it to buckle.