Will there be any problems (I'm talking about copyright...) if we use the lovely smilies of Yahoo! in any of our web apps? eg. forums and so on?
I couldn't find any site that says it is legal/illegal.
Do you know any?
I'm not sure about the copyright laws in your country, but in most countries, especially the ones that abide by the Berne Convention or WIPO Copyright Act and similar international agreements, all creative works are automatically copyrighted whether the author explicitly says so or not.
Aside from fair use, all rights are automatically held by the copyright holder unless they say otherwise. So even if there's no website that says I can't use the 21 Jump Street theme song as the BG music for my Johnny Depp fan site, it'd still be illegal for me to do so because copyrights default to the copyright holder. It's not up to the copyright holder to enumerate all the rights they want to retain. It's just the opposite; they have to enumerate all the rights they want to give up.
So it doesn't matter if they're emoticon graphics or a comic book or a movie; if it's copyrighted (i.e. if it's not in the public domain) then it's illegal for you to violate the copyrights.
In the U.S., copyrighted works don't fall into the public domain for something like 120 years. Though some authors may choose to release a work into the public domain before then, or they may retain the copyrights but use a copyleft license, e.g. the GPL. Or the author may release the work under an open source license such as Creative Commons or the BSD license.
But short of these exceptions, it's not legal to copy a copyrighted work in its entirety for most purposes (there are certain fair use exceptions for teachers, but these are corner cases).
If you want to use the Yahoo! emoticons, then you need to get permission from Yahoo! to do so. Otherwise, you're only licensed to use them as they exist in Yahoo! Messenger.