What happens to the argon in the Haber-Bosch process?

The short answer is that some of the recycle gas is vented at a rate to keep the Ar below about 5% ( I am not a process person so the 5% is a guesstimate ).

Andselisk's answer focuses on the way the ammonia plant can be used to "produce" argon byproduct for sale, but this is by no means essential.

The ammonia reaction cycle is continually purged to remove argon and other undesirable substances. So what do we do if there is no argon plant fitted?

The hydrogen used in an ammonia plant is made by steam reforming. Further H2 is made by the shift reaction:

CH4 + 2H2O --> CO + 2H2  (steam reforming, strongly endothermic, 1000C approx)
CO  + H2O  --> CO2 + H2  (shift reaction, exothermic, 300-400C)

(CO2 removed in older plants by ethanolamine scrubbers, 
and in newer plants by pressure swing absorption)

The steam reforming reaction uses an enormous amount of heat, some of which can be supplied by combustion of process air to generate nitrogen:

(Air) + CH4 --> CO2 + 2 H2O + trace argon + x N2 (x = 8 approx)

However there is insufficient heat from the partial combustion of air to generate all the hydrogen needed to consume the nitrogen produced. Therefore the plant must include a fuel fired furnace using additional air to produce more heat to drive the steam reforming reaction. The exhaust gases of this furnace are sent to atmosphere in the normal way.

The purge steam from the ammonia synthesis loop consists of some argon, with the rest being combustibles or inerts (H2, CH4, N2). If an argon recovery plant is not fitted, the purge gas is simply fed to the furnace and used as a supplementary fuel. The argon is therefore returned to the atmosphere via the chimney.

Inert gaseous components such as methane and argon are indeed should be eliminated from the system in order not to lower the partial pressure of the reactants too much. Technically there is usually a standalone gas separation plant where the extraction of argon from the recycle gas is performed basically using modified Linde process [1 , pp. 428–430].

Suitable cryoprocesses [...] are available for production of noble gases from synthesis purge gas. Initially, extensive separation of nitrogen, argon, and methane is carried out by partial condensation. Purge argon is then recovered from the condensate in a two-stage condensation process. If helium is to be recovered it can be concentrated by liquefaction of the hydrogen in the hydrogen-rich gas phase, followed by purification. The heavier noble gases, krypton and xenon, pass into the methane fraction.

The purge gas at pressures of up to $pu{70 bar}$ is first introduced into the adsorber (a) where traces of water and ammonia are removed. It is then transferred to the heat exchangers (b1) and (b2) for cooling. The gas is then fed into separators (c1) and (c2) where separation of liquefied nitrogen, argon, and methane from gaseous hydrogen takes place, and dissolved hydrogen flashes off. The liquid bottom product is fed into the fractionating column (d1) where methane (bottom) is separated from the nitrogen fraction (top).

A methane-free nitrogen – argon mixture (liquid) is withdrawn from the middle of this column and fed as reflux into the $ce{Ar}$ purification column (d2), where nitrogen is separated from argon. The bottom product is argon of product purity and is transferred to a vacuum-insulated storage tank.

Both columns (d1) and (d2) are operated within an $ce{N2}$ cycle in which cold is produced by expanding high- or medium-pressure nitrogen.

The $ce{CH4}$ bottom stream from (d1), compressed by liquid pump (g), is evaporated against feed gas and normally led to battery limits as fuel gas. The higher boiling noble gases krypton and xenon are contained in this stream and, in principle, can be isolated.

Reference

1. Häussinger, P.; Glatthaar, R.; Rhode, W.; Kick, H.; Benkmann, C.; Weber, J.; Wunschel, H.-J.; Stenke, V.; Leicht, E.; Stenger, H. In Ullmann’s Encyclopedia of Industrial Chemistry; Wiley-VCH Verlag GmbH & Co. KGaA, Ed.: Weinheim, Germany, 2001. DOI: 10.1002/14356007.a17_485.