Am currently using an EF70-300mm f/4-5.6L IS USM L zoom lens on my Canon EOS 450D camera. The zoom lens is used for bird photography and I am thinking of upgrading to a Canon 5D MarkIII full frame sensor camera. Will the lens work efficiently on the 5D or are there any disadvantages as it has been suggested to me there could be with such a full frame sensor camera? I believed any EF lens would be compatible but the rumour has me concerned.
Yes, that lens will work fine on a 5D (or any other full-frame Canon camera). The only Canon lenses which are incompatible with full-frame sensors are the EF-S range.
You will notice a change in zoom factor though. The 1.6x smaller sensor on your 450D means that the lens is currently giving you a zoom range equivalent to 112-480mm. On a full frame camera you'll notice it really is 70-300mm - things won't look as close when you zoom in. However, the massive difference in resolution (12mp on the 450D vs. 22mp on the 5DmkIII) means you can (if you want) perform the 1.6x crop in post-production without losing too much image resolution.
The lens is compatible (any EF lens is compatible with the Canon Full Frame Camera's I believe) however you may get a different "relative" zoom behaviour from the lens on the Full Frame.
the zoom measurement 70-300mm is measured relative to a full frame sensor (or 35 mm film).
However when you are using a non-full frame camera you will get a crop factor by focusing the same lens image on the smaller sensor. (e.g. your 450D has APS-c sensor which is about 1.6 times smaller than full frame)
This essentially means that your 70-300mm lens has essentially been working for you as a ~112~480mm lens (multiply each by 1.6)
So while the lens will funciton perfectly well... you'll probably find you're not getting the same zoom/range out of it.