I'll try to cover all possibilities, as many of the answers here are a bit scattergun and may work, or may not with no suggestion that it may have been the wrong approach for the situation.
Firstly, does it leak with no disconnect attached? If so it is the seals inside the post. There are two, the "poppet" seal and the dip-tube seal. It cannot be the "poppet" seal because it leaks with disconnect attached too (and the disconnect forces the poppet open anyway) and leaky poppets are most obvious because it is plain to see. The seal under the dip-tube is a terrible bit of design, as "Foxbat" says, it needs to be tightened "just right"; too much crushes the O-ring rendering it useless (maybe not immediately), too little and the O-ring doesn't deform "laterally" (outwards as well as up/downwards) and beer/gas leaks through the post threads - this situation can be fixed with PTFE tape on the threads the post attaches to, but the dip-tube O-ring really needs replacing.
Secondly, does it only leak with the disconnect attached? If so it is the seal on the outside of the post. Solution is to replace the seal (O-ring) which can be done without dismantling the post or emptying the keg. "Chesters-mild" did suggest wrapping PTFE tape about the (outside of the) post which may create a temporary seal. Also check that there isn't damage or distortion (or debris) on the sealing faces of the disconnect.
Thirdly, it's what is attached to the post that is leaking. But that should be easy to diagnose, and you've already replaced the tap so it isn't that.
Watch out dismantling the posts. I've not long trashed a thread on a keg that a post screws on to. That's pretty much "bye-bye keg". The O-rings also need to be supple so should be lubricated with whatever is suitable.