Well the most obvious is, the cab needs to be wide enough for the tube amp's feet.
Beyond that, tube amps in general behave differently into a speaker's impedance curve than solid state amps do. A solid state amp approximates a
constant voltage source, so it tries to deliver the same voltage regardless of the speaker's impedance curve. So if a solid state amp delivers say 200 watts into 8 ohms, it will deliver 100 watts into 16 ohms, and it will
try to deliver 400 watts into 4 ohms and 800 watts into 2 ohms... but usually it hits some limiting factor first (usually it lacks the ability to deliver enough current to continue doubling its wattage each time the impedance is cut in half).
A tube amp usually behaves more like a
constant power source, particularly if it has low amounts of global negative feedback (which normally results in a high output impedance, or low damping factor, and a really nice-sounding distortion profile). So this type of amp tries to deliver approximately the same power regardless of the speaker's impedance. So if a tube amp delivers 200 watts into 8 ohms, it will probably deliver close to 200 watts into 16 ohms, and close to 200 watts into 4 ohms.
So now let's consider a hypothetical speaker impedance curve, and look at how these different types of amps behave into it. This is a two-way bass-reflex speaker tuned to about 50 Hz, so it's in the ballpark for what we might encounter in a bass cab:
Let's assume this speaker was "voiced" for a solid state amp... most are. So if we use a tube amp, we'll be getting more wattage into the impedance peaks than the designer expected. So we will have a more upper-mid emphasis because more wattage than the designer planned on will be going into that 1.5 kHz peak; the upper bass will be louder (which may or may not be a good thing) because of the impedance peak at 70 Hz; and since the peak at 30 Hz is below the tuning frequency, it will tend to make the cab a bit more likely to fartout on a low-B than if we used a solid state amp. This speaker may very well sound better with a solid state amp than with a tube amp, unless it just so happens that the changes are improvements to our ears.
Let's look at another impedance curve, this time just a woofer (no mid or tweet) in a sealed box, like an "old school" bass cab:
Relative to a solid state amp, our tube amp will put more wattage into the speaker at the 45 Hz impedance peak, and it will also put out increasing wattage as we go up in frequency. Now more than likely a sealed-box bass cab would have it sbass peak in the 60-to-80 Hz region, so that is where the low-end "boost" would happen. So this speaker will have more bottom end, as well as more top end,when driven by a tube amp - and since both are usually desirable in a bass cab, it's not surprising that "old school" bass cabs tend to sound better with tube amps.
In the real world the differences may or may not be as dramatic as I have described here, depending on the specifics of the amplifiers and speakers involved. But the principles described are correct to the best of my knowledge and according to measurements I've made.
For a more thorough look at the subject of different amplifier paradigms, you might take a look at this paper:
Paradigms in Amplifier Design
That was probably a longer answer than you were looking for. Sometimes my inner geek just takes over.