I have read that the Arabic maqam system has 24 pitches in an octave. Are there any instruments (keyboards especially) that have been made specifically to be able to play all 24 tones?
First of all, let's get this fact out of the way: 24 pitches in an octave is a rather simplistic notational convention. It doesn't reflect the actual state of affairs at all. Actual number of notes can be much higher. One practitioner, for instance, claims to have identified at least 12 notes between his lowest e-flat and his highest e-natural. Anyone who tries to perform Arab music (or the music of other closely related traditions, like Turkish) with 24 equal-tempered tuning will quickly realize that it sounds horribly out of tune (for both western and oriental ears).
But this is for all possible maqams. In a usual musical piece, one maqam will be used, often with occasional modulations to other related maqams. The number of notes in a single maqam is usually only slightly higher than 7 per octave. So, in a usual song, you will most probably be using fewer than 12 notes per octave.
In some electronic keyboards, individual notes can be microtuned. This way you can play, for example, in the maqam rast on C by slightly lowering your E key (and with smaller slight adjustments to other keys). Such keyboards usually offer a preset system for tunings too. So you can change to appropriate tuning for the maqam before each song. From what I've seen, this seems to be the most popular choice for performing maqam music on a keyboard.
Keyboards with more than 12 keys per octave are uncommon but not at all unheard of. You can find literally hundreds of designs by searching for "microtonal keyboard". There are also solutions that sense which region of the key you press to change the pitch accordingly.
For other instruments, authentic Arab (and Turkish) instruments are of course capable of producing more than 12 notes per octave:
For string instruments, one obvious solution is to go fretless, like in oud. Turkish tambur uses 21 to 35 (depending on the practitioner) movable frets per octave. So, the performer can make slight adjustments in between pieces in order to better accommodate the maqam of the piece.
Turkish qanuns have small levers for fine tuning that enable the performer to divide the octave in roughly 72 equally spaced pitches. Here you can see the performer manipulating the levers with his left hand. That's right, seventy two!¹ Take that 24 :)
Apart from that, several western instruments can also produce microtones. Clarinet (sometimes with slight modifications) is actually quite popular in Turkish music for example.
Finally, there are experimental designs that are yet to prove popular.
¹ Actually that's not quite true. Usually not all 72 pitches are available.