Bitcoin will change nature from inflationary to deflationary currency. This is as designed from the very beginning - there is an upper limit of how many bitcoins can there be. This is actually profitable for the stability of the currency - now it's prone to rapid exchange rate changes if someone starts a big mining farm and introduces a lot of it to the market. In the future this will not be the case - with fixed number of coins the value will only depend on supply and demand based on exchange of assets, not on "side stream" of "mining".
With coins falling out of circulation by being frozen as savings the number in circulation will be constantly falling.
Of course falling number will cause increased value of single coins and as result encourage people to spend their savings and reintroduce frozen assets into circulation, so the time before there's not enough bitcoin in circulation to support continued existence of the currency is very distant and we're likely to develop some alternatives before this happens.
Of course while at certain point continued purchases of mining hardware may cease to be profitable, running existing hardware is cheap, and so mining is unlikely to cease for quite a few years yet despite its dwindling profitability.
When cost of electricity to mine one bitcoin will exceed the value of one bitcoin you may be sure production will stop (and it will be long before you need a whole warehouse of dedicated hardware to produce one.) Of course there will be people still producing them at a loss or using public/stolen/free electricity but they won't produce all that many of them.
Now let's assume the unlikely "too long chain" scenario eventually happens requiring a whole warehouse to process - which shouldn't happen considering the upper cap on the number of coins, unless someone miscalculated something - last bitcoin should be eventually generated on much less beefy hardware. Once you reach that limit, you will be unable to mine another coin, and so you won't obtain it, period.
If a bitcoin is already mined, it really doesn't put much stress on the infrastructure, it requires much, much less processing power to transfer as a payment or validate than to create. It's the mining process that would eventually grind to a halt first ages before the processing of mined bitcoins would become too hard on hardware.
If you had a warehouse of custom FPGA, you'll be left with a ton of useless mining hardware and only as many bitcoins as you obtained. The era of mining will come to an end as designed, and the bitcoins produced from even the longest chains will function just like any other ones in circulation.