In a trade imbalance does the importing country's currency become more valuable? Suppose that country A buys freely from country B, but B uses steep tarriffs to block its citizens from buying from A. In that case country A's currency will pile up in B.

Other things being equal, will this cause currency A to become more and more valuable relative to currency B?
No. In fact the opposite happens. The value of a currency (or the exchange rate) is determined by supply and demand of the currency on the currency market. If country A imports more, the supply of country A's currency will increase more, therefore losing its value.

Suppose B's goods are priced in B's currency. Then, the supply of A's currency increases in your example, because it needs to sell its own currency to purchase more of B's currency to be able to buy B's goods. Therefore more of A's currency comes on the market, while more of B's is demanded. If priced in A's currency, the exporters in B will still want to have B's currency as they need it because they live in B. Then they will want to exchange the amounts of A currency obtained for B currency.

Hence, in both cases on the markets for currency, demand for B's currency increases, while supply for A's currency increases, therefore reducing the value of A's currency.

The currency would not (necessarily) "stockpile" however, as import tariffs in B do not influence the exchange of currency.