The Toffees, whose appeal hearing over their 10-point deduction for breaching financial rules begins on Wednesday, slip to 18th, a point behind Luton following the Hatters' win over Brighton.
Despite the scoreline there was no shortage of goalmouth action at Craven Cottage with both sides hitting the woodwork and having efforts cleared off the line.
Everton endured both in a matter of seconds as James Tarkowski's scuffed shot flicked the head of Issa Diop and looped up to hit the crossbar. The ball then brushed Antonee Robinson and rolled back towards the goal, only for Timothy Castagne to poke it away in the nick of time.
Castagne was on the receiving end of a goalline clearance later in the first half when his fierce shot was deflected past Jordan Pickford and Tarkowski was on hand to block.
After a fairly even first half, Fulham dominated the second with Castagne's flicked header coming back off the bar and Pickford making a stunning save to keep out a Tosin Adarabioyo header that looked destined to find the net.
Marco Silva's side kept their visitors penned inside their own half for much of the second period but failed to make the most of numerous promising situations and were nearly punished.
Everton substitute Beto headed over from point-bank range in injury time and the Toffees were unable to force the ball home during a scramble on the edge of the Fulham six-yard box a minute later.
To add to Fulham's frustrations, Raul Jimenez went off holding his hamstring at the end of the first 45 minutes and had to be replaced at half-time, with Diop also grabbing the back of his thigh after being substituted in the second half.
Familiar frustration for Fulham
It is now five games without a win in all competitions for Fulham after they were unable to put the finishing touches to an encouraging display.
The Cottagers have failed to score in a league-high 10 games this season and despite having 25 shots their profligacy in front of goal remained.
Jimenez was missed as they pushed to break the deadlock in the second half but he was guilty of passing up a very good opportunity when he sent his shot spinning wide after Castagne's effort was cleared off the line.
Their lack of ruthlessness in front of goal was exemplified on the hour mark as Bobby Decordova-Reid broke free after winning a 50-50 challenge with Jarrad Branthwaite.
Willian was up in support but a superb two-on-one situation did not even result in a shot as Decordova-Reid delayed the pass just long enough so that shooting became the better option, and Ben Godfrey was able to cut it out.
"I have to say the goals are going to come. We cannot keep creating chances and not score," Silva told BBC Match of the Day.
"The decision-making has to be better and be calm enough to make the right decisions. The decision-making is having a big impact when you arrive in so many dangerous situations and we have to do better to score more goals."
Silva's team should not be involved in a relegation scrap this season but will know that until they can become more clinical, they risk being dragged in.
A point gained and two dropped for Everton?
Given the way the second half played out, Everton would surely have happily accepted a point when the board went up to indicate six added minutes.
However, given the glaring nature of Beto's miss and the opportunities for the Portuguese forward and Godfrey in the goalmouth chaos that came soon after, Sean Dyche will be wondering how his side managed not to win.
It would have been a significant boost at the start of a crucial week for the club as their appeal against a points deduction gets under way.
That they now find themselves in the relegation zone only serves to highlight how much is riding on the outcome.
Dyche was able to use the initial punishment as a way of galvanising his players but after a good spell before Christmas their form has dipped, albeit during a tough run of fixtures, and they were distinctly second-best after half-time in west London.
"We're a bit stretched at the moment, but there was maximum effort and maximum mentality," Dyche told Match of the Day.
"A good 0-0, if there is such a thing, but we're stretched and asking a lot of them, even from players that are not totally match fit.
"The group are giving everything, supported by a fantastic fanbase. There is a lot of work to do but clear signs of a group giving everything for the cause, even when we have a lot of injuries."
Every point counts and this one could make all the difference come May, they can only hope that they aren't looking back and seeing this as a vital two points dropped.