War is fought with money. It was true in the old ages, and even more true in Victoria 3. Aside from money, however, are the allies willing to take that money from you in exchange for goods.
A very tangible cost is of course the actual money you are spending on military goods for mobilized troops in the field. While Barracks consume military goods at all times in proportion to the number of Battalions it supports, units that are mobilized consume twice the normal maintenance level of goods.
Not only does this double the quantity you need to buy, but it also puts a lot of additional demand on the goods, raising the price-per-unit as well. As long as they can keep their production levels steady, your domestic Arms Industries, Munition Plants, and War Machine Industries will see profits skyrocket off of this increased consumption of your mobilized troops.
This also means other nations - allies or neutral parties - will see the sudden benefit of exporting arms to you. If you normally export military goods to others you may wish to cancel these routes to keep prices down, while if neutral parties import arms from you this may be a good time to Embargo (or at least Tariff) these goods. Every little bit helps your treasury and ultimately your frontline troops.
Many countries rely on foreign trade to supply them not only with the additional arms they require during wartime, but also the consumer goods required to keep morale up on the home front in this difficult time. Ending up on the opposite side of your trading partners during Diplomatic Plays could be catastrophic for the war effort.