Is there any function in LO to increase/decrease the font size via a shortcut or a button? How can I define it and assign it to a shortcut or a button?
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This explanation is for the Writer in version 4.0.2 (and, parenthetically, whether they apply to Calc). I am not that familiar with other applications in LibeOffice to be able to comment about them.
It looks like the default keystrokes for increasing the font size and decreasing the font size are Ctrl+] and Ctrl+[ respectively. (Does not apply to Calc.)
On the formatting toolbar, one typically has (among other items) the name of the style to be applied, the current font name, the font size, then icons for Bold, Italic, **U**nderline, … One can use the pull-down for the font size to change the size of the font. (This also applies to Calc.)
One can also use the pull-down for the style (and even customize a style) that sets a different font size. (Also applies to Calc.)
The icons for increasing & decreasing the font size, A^ and Av are apparently by default not displayed. If you right+click anywhere on the Format toolbar, and then select “Customize toolbar…”, in the “Commands” window is a long list of formatting commands with a place to check in front of each one. About 80% the way down the list are places where you can check for the icons for increasing and decreasing the font size. You can also change where they appear on the toolbar once they are selected. (The “customize toolbar” option for Calc doesn’t seem to have an increase font size / decrease font size icons to select.)
One can also define additional keystrokes in the Writer for increasing and decreasing the font size by going into Tools | Customize, picking the “Keyboard” tab, in the top half (“shortcut keys”) picking a keystroke you want to have a new action, and in the bottom (“functions”) picking the action (e.g., Format | Reduce Font, or Format | Increase Font). When the keystroke in the top half of the window is matched with the desired action in the bottom half of the window, click the “Modify” button. (Does not apply to Calc because “Increase Font” and “Decrease Font” are not Format options in Calc.) Unfortunately, one cannot use Ctrl+Shift+< and Ctrl+Shift+>, or at least Ctrl+Shift+<, because the keystrokes involving < (the “,” key) are omitted from the list of possible keystrokes.
(Calc has “Format” | “Cells” or hotkey Ctrl+1, and then on the “Font” tab one can pick the size.)
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In Writer and Impress 3.6 you can customize a toolbar by adding a button increase and another to decrease the font size. I assume the same is possible for Draw and Calc.
The default short cut keys are the same as described in @oweng’s answer. Most if not all what he describes is the same in 3.6.
Update 2014.11.03 - This “bug” (fdo#35862) has now been “fixed”, and will be available in 4.4.0 according to the note on Bugzilla. That’s good news! | 2015.02.01 I just got 4.4.0.3 through PPA update, and can confirm this is now implemented. Nice. 
This extra answer is simply to point out a limitation of LibO Writer that will be unexpected to those accustomed to the way this works in MS Word.
In Word, CTRL + ] increases the font size, whatever it is, by 1; conversely, CTRL + [ decreases the font size, whatever it is, by 1. In mixed size text, everything either increases or decreases by 1 point at a time.
This is not the behaviour of Writer. For one thing, it moves in 2-point jumps rather than 1. But more importantly, if you select text of a common size, you can use the shortcuts to increase/decrease:
However, in selections of mixed-size text, this option is not available:
As I say, those familiar with this feature in Word will miss it in Writer – it’s probably worth an enhancement request (P.s. now submitted: fdo#79796).
(As I post this answer, Writer is at 4.2.4.2, stable release.)