Boiling point trend in group 13
London forces are the forces that you should always consider in the end. They're like the slow pawns on the chessboard, their effect is negligible when the fast rooks and the industrious queens are around!

In this case, our queens are the different structures of these elements. Let me first tell you the fact that even the melting point of boron ($\pu{2349K}$) is more than the boiling points of thallium and indium!

So, first off, why is the melting point of boron higher than that of all other group thirteen metals? From this Wikipedia page, its α-rhombohedral allotrope has B-12 icosahedral units arranged in a crystalline structure, and connected by strong covalent bonds[a]. As a comparison, the other metals in their solid state have unit cells whose atoms are bonded by only weak metallic bonding. The larger the atoms get, the weaker the metallic bonding gets, and hence, the melting point decreases down the group.[b]

Coming to the trend in boiling points now. Part of the difference in the trend has to arise from the excessively large melting point of boron. You see, thallium boils off even when boron has yet to melt, so that is a contributing factor in the excessively large boiling point of boron over the others.

According to this paper[1], for liquid thallium,


  The coordination number changes from 11.5 atoms at $\pu{350^\circ C}$ to 10.9 atoms at $\pu{700^\circ C}$


This is said to be due to:


  creation of vacant spaces about an atom. Although in liquid thallium the interatomic distances remain unchanged, some of the atoms are moving farther apart within the first coordination shell.


Another paper[2] however shows that, while CN of liquid indium decreases with increasing temperature, their inter-atomic separation also decreases. We can interpret this to be a "reinforcement" that helps increase the boiling point of indium over thallium.

With indium and thallium cleared, notice that the boiling points of indium, gallium, and aluminum are almost identical. Sadly, I've been unable to find any research papers dealing with these three elements in liquid state near their boiling point. For example, this paper[3] deals with liquid aluminium, but only upto $\pu{1273K}$.

We do have some data for boron. You'll be surprised how strong its covalent bonds are. In fact, this paper[4] states that, even at as absurdly high temperatures as $\pu{2600K}$, though the "icosahedra are destroyed", the "atoms still form an open packing with sixfold coordination". Moreover, this paper[5] observed that the density of liquid boron, at temperatures above $\pu{2360K}$, is higher than that of solid boron, which is unlike aluminum and indium (though for gallium the density at melting point is still slightly higher). Another paper[6] has measured the properties of liquid boron, but that sadly only at around $\pu{2400K}$.



I have checked a dozen best-matching research papers, and this was all the relevant data I could pull out. This isn't the complete answer, but again, without any proper data in the first place, there really cannot be a proper qualitative analysis either. So, this answer is really all there is.



References:


Temperature Dependence of the Structure and Transport Properties of Liquid Thallium, N. C. Halder and C. N. J. Wagner, The Journal of Chemical Physics 45, 482 (1966); https://doi.org/10.1063/1.1727593
Temperature Dependence of the Structure of Liquid Indium, H. Ocken and C. N. J. Wagner, Phys. Rev. 149, 122 – Published 9 September 1966
The structures of liquid mercury and liquid aluminum, P. J. Black and J. A. Cundall, Acta Cryst. (1965). 19, 807-814 https://doi.org/10.1107/S0365110X65004383
Structural and electronic properties of liquid boron from a molecular-dynamics simulation; N. Vast, S. Bernard, and G. Zerah; Phys. Rev. B 52, 4123 – Published 1 August 1995
Millot, F., Rifflet, J.C., Sarou-Kanian, V. et al. International Journal of Thermophysics (2002) 23: 1185. https://doi.org/10.1023/A:1019836102776
Structure of Liquid Boron; S. Krishnan, S. Ansell, J. J. Felten, K. J. Volin, and D. L. Price; Phys. Rev. Lett. 81, 586 – Published 20 July 1998; https://doi.org/10.1103/PhysRevLett.81.586




[a]: actually "the icosahedra are connected via a combination of two-center and three-center bonds" as claimed in [6], but suffice to say that they are almost as strong as the covalent bonds, if not more
[b]: except a few exceptions