Lets start with some caveats, PvP interactions in D&D 4e are poorly defined and effectively mechanized. The second part is that your group's social contract should include the expectations for PvP conflict, the resolution method and whether or not PvP combat is allowed (among other things, there are some good answers about social contracts here if you need more info).
Now, in 4e Diplomacy is always opposed by a DC. However, the DC should be set based on the character's attitude, number of characters influenced and temporary modifiers that depend on what is trying to be accomplished. A suggestion for allowing the opposing player to be involved is to set the DC (a bit lower than you would normally), and allow the Player to roll a check (maybe a Wisdom/CHA Check?) to add to the DC (maybe at 1/2 check).
PvP makes it a bit more difficult to decide a DC. However the amount of leeway given to the DM should allow the DM to decide (based on intended plot) how to set the DC so that either a success or failure is meaningful.
Lastly, I really want to ask (and I wonder this in my group when these situations come up), what purpose does this conflict serve to the narrative? Is it just petty infighting amongst PCs/Players? or does it serve a narrative purpose? If it's enhancing the roleplaying experience for the players than that's great, but if it's just frustrating to all involved then narrate it an move on.
A few games resolve this situation by dealing with it explicitly in the rules, and building the check system to accomodate how it handles this situation. The most notable one is Burning Wheel and its Let It Ride rule:
The result of one test stands for the duration of the situation.
When in a situation like this, the success or failure of the attempt comes down to one roll. Players may cooperate or not, but once the dice are cast for the first time, the door will/will not be open and there can be no new roll by anyone for that goal until the situation significantly changes (like, they go fetch a battering ram, or they return a month later).
Lots of Burning Wheel's rules are tightly enmeshed in the rest of the rules, but this is one of the few that is easily separated and portable to other games. If your game of choice already has rules for assisting, you don't even need to houserule anything once you let your players know that you'll be following Let It Ride from now on. In the situation you describe, the fiction doesn't even need to change. "We all take turns bashing at the door. Eventually one of us must get through!" After establishing their method and their goal, they decide who's making the roll, everyone else adds bonuses for helping, and then the one roll is made.
This rule was built into Burning Wheel specifically because the author got the same feeling as you, that there's a problem with this common occurrence in games with skill systems. Most games test for the action, which under some circumstances virtually guarantees success (or failure) by just repeating the action. Instead of using task-based resolution like that for skills, Burning Wheel uses an intent-based resolution, where your goal is why you roll (and only once), but what you roll is determined by the method you use. Let It Ride is a key part of its intent-based resolution, in that it reminds everyone that they only get to do this once, so they need to bring all their resources to bear – or not, if someone in the party is opposed to the attempt. Either way, everyone has to commit to either pursuing this goal or not, before the roll happens. There's no hanging back to try after, because there's no second chance.
As an added bonus, it fixes a related problem, because the rule binds the GM too. The GM is forbidden from calling for multiple tests for the same task, so there's none of this sort of thing designed to railroad a failure:
GM: Roll to climb! Hm, success… okay, you get 10 feet up without falling. Roll again! Another success… What's your skill level? Oh, that's pretty good… So you reach a ledge 30' up. You've got 100' to go, so roll again! …Success. You get halfway up. It's a long way down now. Roll climb. Aw, you failed! You plummet to your death. But wait! Giant eagles save you and fly you halfway around the world.
Best Answer
Introduction
As I understand it, you and your players are largely okay with the fact that this guy has incredible skills. Your problem is specifically that you want the other players to have a reason to put skill points into something that this guy can already do pretty well. There are some things you can do to encourage this, but if there is a broader problem of "it's hard to design adventures so that this guy is useful but not game-breaking", that's probably best handled out-of-game at this point.
Redundant Skills Aren't Useless
Some skills are just naturally good to have on more than one character. Some examples:
I'm Just One Man
You can make other skills useful by playing on the fact that the skill monkey can only be in one place at a time. You don't necessarily have to split the party to do this, either:
The beauty of this situation is that the skill monkey is a big help but can't do it alone, whichever role they take. They might be able to search quickly or distract one group of guards with little chance of failure, but the other PCs still need to be able to handle their parts of the plan.
Chase Scenes
Several PFS scenarios feature chase scenes, where a bad guy flees and the PCs can only make up the distance by succeeding at a certain number of rolls, mostly skill checks. Some of those scenarios include:
The skills used in these scenes vary, but usually you have a few different options at each stage. For example, if a crowd is in your way, you might parkour around them with Acrobatics, cut through an alley you know about because of Knowledge (local), or push your way through with a combat maneuver check.
These chases can be win-or-lose - the PCs catch the bad guy or he gets away - but they don't have to be. For example, the bad guy might be running to his gang's hideout. If the PCs beat him there, you can surprise the gang. If they're right behind him, it's a standard combat. If they're lagging behind, the gang might have a few rounds to cast buff spells and activate the pit trap by the door before the PCs arrive.
In-Character Reasons
The fact that the skill monkey is a consummate liar might mean that the other PCs want to do some things to keep tabs on him. Even if the players all trust each other, that doesn't mean that their characters necessarily do. For example: