Which spells benefit from the Spell Sniper feat?

The majority of spells that benefit are those with a ranged attack roll, therefore we can start with all spells that require a ranged attack roll.

Some of the above spells, while requiring ranged spell attacks, do not do so as part of casting and are thus an exception. These are Crown of Stars, Storm Sphere, and Wall of Light. Crown of Stars is an even bigger outlier as it is a spell of range 'self'.

This is because, as Double Range for Storm Sphere with Spell Sniper mentions the benefits of doubling the range come,

When you cast a spell...

Thus from a strict reading of the rules, for all of the above named spells (where the attacks all occur separately from casting), they do not benefit from the range increase. But those ranged spell attacks do benefit from the reduction in cover granted by the feat.

The Produce Flames cantrip is also an outlier, which could arguably be ruled either way. The relevant text is:

You can also attack with the flame, although doing so ends the spell. When you cast this spell, or as an action on a later turn, you can hurl the flame at a creature within 30 feet of you. Make a ranged spell attack. On a hit, the target takes 1d8 fire damage.

Thus, the attack may be done as part of casting the spell, and this attack has a range, even though the spell range is 'self'. The attack being part of the casting distinguishes it from the other outliers above.

On top of the initially linked list of ranged spell attacks, the melee attack spells Booming Blade and Green-flame Blade also benefit, so long as you can attack out to 10ft with your melee weapon (i.e. your weapon has at least 10ft reach).

"When you cast a spell that requires you to make an attack roll, the spell’s range is doubled."

Notice that this doesn't specify ranged attack rolls: don't forget there are several melee spells that require you to make a melee spell attack roll on something within 5 feet of you, instead of at touch range. So if you have spell sniper, this doubles to 10 feet.

You can follow the link in the other answer and sort for melee in an advanced filter to see them, too.