My thinking is that such a war is highly unlikely to come about. At the VERY most, it starts as a series of border incidents following aggressive posturing by both sides - no way is either Mussolini or the French going to willingly and deliberately go against the other.
So, given that, what's the most likely way such a war can develop ?
For one, the British are going to support the French diplomatically and impose sanctions against Italy, claiming they started it (regardless of whether that's true or not). Meanwhile, the French will be constantly looking over their shoulder, checking what the Germans may be up to, fearing an attack whilst their off on wild goose-chases in the African deserts or Alpine heights.
What does this mean ? Well it means both Italy and France have an incentive to finish the war early - France for fear of a German backstab and due to economic concerns (the main reasons why they didn't contest the Rhineland occupation OTL was that it would have triggered a capital flight out of France that would have led to their default within days), whilst Italy due to the adverse effect of the sanctions and fear of a British entry. Therefor, we can safely conclude that such a war is going to be relatively short.
Now about the action:
French doctrine was centered around static defense of fortified positions, whilst Italian doctrine was all about defending Alpine mountain passes and hills, followed by a counterattack.
Just like the French, the Italians also had their own
fortified line along the Alps, and we can safely assume that any French attack in this area, on the off chance that it happens in the first place, is going to be beaten back. Everything - the terrain, the fortifications, the enemy numbers, equipment and doctrine - leads to this conclusion. Likewise for the Italians, on whom we have a historical account of how they fared - after 3 days of fighting (with the main advance held up by a snowstorm), France lost 200 men KIA/MIA and a further 1400 taken prisoner, whilst Italy lost ~ 1200 men KIA/MIA and a further 2600 casualties due to frostbite. So, basically, the Alps are a terrible place to be fighting in, and neither side is going to make any significant gains.
This leaves the other theaters. In East Africa, it's pretty much a given that French Somaliland is going to be taken, and this might well be the only significant gain anyone is going to make.
In the mediterranean islands, nobody really has the numbers or capability to mount a successful amphibious operation.
Regarding naval combat, both countries have similarly sized fleets, and both are known for using them conservatively (especially Italy). I doubt anyone is going to risk their battleships and, if they do, their orders will be such that they'll decline any but the most one-sided of engagements. This means the the brunt of the fighting is going to be carried out by lighter units, and, besides the sporadic raids and counterraids up north, the main objectives are going to be the blockading of Tunis and Tripoli, respectively.
Here, the French are at a disadvantage, since their naval bases at Tunis and Bizerta are very vulnerable to Italian naval and air assets which basically surround them on 3 sides, stationed in Sicilly, Sardinia and Tripoli. It will be much harder for the French to escort merchant vessels into Tunis than it will be for the Italians into Tripoli. Thus, French supplies are going to have to be mostly brought up overland from probably as far away as Algiers, with some of the smaller ports between there and Tunis being either too small, too vulnerable or both. Having to bring their supplies over such a long distance, French commanders are going to be unable to sustain any kind of offensive towards Tripoli for long (not that they would want to attack in any direction anyway), since the large density of troops on both sides (OTL, Italy had 10 divisions sitting at the Tunisian border at the start of the war) would probably require a rate of ammo consumption way above what French supply lines can handle. Moreover, the Italians are fighting in their preferred setting - temperate climate, close to their supply depots, defending a series of hills. The last argument against a French victory, as so many posters seem to fantasize about, is that the French have ZERO experience fighting in such a setting, whilst the Italians can bring up all of their veteran from the Abissinian campaign.
OTOH, if Mussolini will want an offensive into Tunisia, he's going to be severely disappointed, as the French positions along the Mareth Line, again filled to the brim with French soldiers, are going to thwart any attack.
Thus, apart from Djibouti, neither side is going to capture anything of note, and both will be eager to have this stupid war (they didn't want in the first place) over.