Can I change from short to long derailleur to increase cog size

You may need a longer derailleur, because you're going to need to run a longer chain to compensate for the extra teeth. Try swapping the chain first to see if the new cassette work with the shorter derailleur arm, but you may find that the chain will be too long, sag, and rub on itself between the pulley cogs when you're riding on your smallest ring. Whether it'll work or not depends entirely on gear ratios.

It doesn't look like you'd be able to use an 11-32T cassette at all with a 10-speed Tiagra rear derailleur. From [Shimano's documentation] (http://bike.shimano.com/media/techdocs/content/cycle/SI/SI_5XN0A/SI-5XN0A-001-Eng_v1_m56577569830746860.pdf), the largest sprocket that either the short cage or long cage version supports is 30T. If you wanted to use their 12-30T tiagra cassette, you would need the long cage because the tooth difference has to be 16 teeth or less for the short cage, whereas the long cage supports a tooth difference up to 22 teeth.

If you really want to go up to 32T and stick with a road derailleur, it looks like you'll have to upgrade to the newer 11 speed stuff and get the long cage version. The [11 speed Ultegra] (http://bike.shimano.com/content/sac-bike/en/home/road/shifting---braking/rear-derailleurs/rd-6800-gs.html) derailleur looks like the only road derailleur from Shimano that would support an 11-32T cassette. Incidentally, that's the derailleur that came stock on my 'cross bike and it's got an 11-32T cassette on there.

The short and long deraileur cages each have a specified "capacity". This capacity is the sum of the (largest cog - smalest cog) + (largest chainring - smallest chainring).

For example, if you have a tripple 30/39/52 tripple with that 11-32 cassette, you will need a deraileur capacity of (52-30)+(32-11)=43. I don't think they make them that large - current Tiagara large is 39, but check on your model/year specifically. With a 39 capacity deraileur cage, you can afford only 11 difference on the front, will only work with a 36-46 cyclocross style crank.

As the other answer stated, you will likely also need a new chain.

That being said, while it is not recommended, you could use a derailleur cage that falls a bit short on capacity as long as you avoid the small/small gear combos. If you go this way, first make sure your chain is long enough to accommodate the large/large gear. Then make sure you know which small/small gearcombos are going to be off limits by testing before riding the bike (or calculating from capacity), and make sure you NEVER try to use those gears. Make sure you understand all this before proceeding, it is as I said not recommended.

I believe when the pros pull tricks like this for especially steep mountain stages, they are using a MTB derailleur or the cage from a MTB derailleur on their road derailleur - or the rider is just really confident he knows what he is doing.

I'm running a 46/34 and 11/34 with a 105 short cage with no problems. The LBS was surprised that it could be done. I didn't know any better so I did it.