I am sorry if this is obvious but I want to be sure. Will 1 monitor of a high resolution (2560x1440) use more graphics power than a monitor of a lower resolution (1920x1200)? Will the increase in 'graphics power used' be noticable?
yes, depending on what you use the pc for.
for example, playing a game at 2560x over 1920x can drop your framerate by about half. using a 2560x monitor will require you to spend more on a gpu and update the gpu more frequently if you want to keep the same quality settings and fps.
also keep in mind that movies are 1920x1080 so if you plan on watching movies this is the resolution you want. on 2560x you might see a loss of quality. this applies to other pre rendered material as well. for games and other applications which render on the spot 2560x will be about the same visual quality as 1080p viewed at the same distance. what you gain is not sharpness but overall size since the actual pixels per inch are actually a bit lower.
Your graphics card will give each monitor it's chosen resolution and there is no noticeable difference in the use of two different resolutions. I'm sure that if there was a way to play an intense game on each monitor seperatly that you may notice the difference and that difference would be in the heat generated by the card.
Resolution is the drawing of lines of the pitcure to be displayed and the 2560x1440 will have more lines then the 1920x1080 so it's more time consuming the power consuming , however most modern video cards support 2560x1600 so they are more then capable of doing the two resolutions that you stated.
of course both resolutions are supported. what the op is asking is if there is a performance decrease by increasing the resolution.
the answer to this would be yes.
look at the toms hardware charts. specifically the game frame per second charts. look at the difference between 2560 and 1920 it can be pretty significant on cutting edge games while not as significant on older games.