It is possible that web pages intercept the left mouse button via JavaScript and prevent opening the link in a new tab.
If your mouse has a scroll button then middle-click the link instead.

Edit: You might already have this set if the right-click option always works.

By default, Firefox will divert standard new windows to a tab, but if the link actually is a script that launches a window with special parameters, like a dialog, then Firefox will honor that. There is an obscure setting to disregard those parameters and force all scripted windows into a new tab as well.

(1) In a new tab, type or paste about:config in the address bar and press Enter. Click the button promising to be careful.

(2) In the filter box, type or paste link.o and pause while the list is filtered

(3) Double click these and set the value as desired:

(A) browser.link.open_newwindow

3 = divert new window to a new tab (default) (checked*)
2 = allow link to open a new window (unchecked*)
1 = force new window into same tab

* First checkbox in Options > Tabs

(B) browser.link.open_newwindow.restriction

0 = apply the setting under (A) to ALL new windows (even script windows)
2 = apply the setting under (A) to normal windows, but NOT to script windows with features (default)
1 = override the setting under (A) and always use new windows

Does that work?