Can tricep pain be cause by having a smaller bike frame?
NOTE that this answer is looking at the OP's question of fit (bike sizing, frame geometry and body proportions.  
  
  Other answers already exist that look just at tricep pain: Symptoms of handlebars that are too narrow? Triceps pain?


Fit is really hard to diagnose without directly observing.  On paper a 55 cm is about the right size for someone who is 178 cm. This is of course on average (average bike, average person).

Unfortunately, no bike or person is this mythical average.  Some bikes are more racy and will have a longer reach and lower stack height for a "55 cm". Others could be the opposite (taller stack, shorter reach) if they are targeted at more endurance riding.   

People differ as well.  For someone with long legs and a short torso fit can be harder, you generally need a taller stack and a shorter reach, as your long legs put the saddle up higher (relative to someone with shorter legs) and therefore your handle bars tend to be relatively lower and longer away for any given frame geometry. The Kestrel Talon is a Triathalon bike with a frame geometry emphasizing a shorter stack and long reach (i.e., race position). This is probably one of the worst frame geometries for someone with long legs, a short torso, and poor flexibility.  If you boyfriend is quite limited in his flexibility, it is possible that no Kestrel Talon frame sizing would provide a good fit (e.g., you could go a size up to get a higher stack height, but this also adds to the reach, which can be problematic for someone with a short torso).

Assuming we don't have an extreme fit mismatch, there are things that can be done to modify the fit.  Start by putting all the spacers under the stem and even flipping the stem up so that it becomes a riser stem. There are also a lot of aftermarket stems available that can further accentuate the effect. Stem length can be used to dial in the fit, as well as the handle bars which also have different reaches.  

Saddle setup can also impact tricep use, if the saddle is tilted nose down the rider may continually slide forward, using their arms and tricepts to push themselves back into position on the saddle.

Handlebar width can also affect tricep fatigue:


  It's important to get this measurement correct as too wide a hand placement leads to fatigue and numbness in the hands, due to their [sic] being splayed out.  This also affects handling, making turning the bike slower. Having too narrow a hand placement can be tiring for the triceps which have to bear a greater load, and will affect the handling by making the steering quicker and the bike 'twitcher'.
  
  -- Bike Fit by Phil Burton


Finally, flexibility and core strength also has a huge impact on road bike fit. You need good flexibility in your hamstrings, hips and mid back if you want to comfortably get long and low (i.e., a more "race" position like the Kestrel Talon). If you lack flexibility or strength you will find that your body is forced to compensate in different ways.  Because the body is very dynamic, different people will often compensate in different ways, resulting in different symptoms.  Tired triceps could be one sign that the handle bars are too low, or that your boyfriend cannot sufficiently support his upper body weight with is core muscles.  Riding low and long (which the Kestrel Talon geometry generally enforces) requires good core strength and more intense pacing. If you lack any of the three (core strength, flexibility, and intensity) you will need to find a handle bar position that is higher and closer to find that same balanced position. Ideally, when you hit the right fit for your intensity, flexibility and strength you should feel that your hands are lightly touching the handle bars under power.