They won't need sleep, but they will need some form of rest to regain daily spells.
While fused with their eidolon, a Synthesist summoner gains the benefits of its evolutions:
The synthesist also gains access to the eidolon’s special abilities and the eidolon’s evolutions.
Because of the wording of "base evolutions" in the unchained eidolon's features, their progressive upgrades also qualify as evolutions. The 4th-level unchained eidolon of the Inevitable subtype gains an evolution that grants immunity to fatigue and exhaustion. And at 12th level, they gain immunity to sleep. Not just magic sleep.
At 4th level, inevitable eidolons gain a +4 bonus on saving throws against mind-affecting effects and immunity to nonlethal damage, fatigue, and exhaustion.
At 12th level, inevitable eidolons gain DR 5/chaotic. They also gain immunity to sleep, as well as the true speech ability (see the inevitable subtype).
So the synthesist (while merged with their eidolon) can skip sleep without becoming fatigued. And at level 12, they gain sleep immunity, and can't be put to sleep.
However, this does not affect the rest time required for the purpose of reading their daily spells. Whether or not a spontaneous caster is fatigued or exhausted, they still require rest and concentration in order to get their spells ready.
The rules for arcane spellcasting were originally published in the Core Rulebook, when the only arcane spellcasters were sorcerers and bards. But these restrictions also apply to other arcane spellcasters, which includes the unchained summoner. The quoted rules below should therefore apply to summoners.
Each day, sorcerers and bards must focus their minds on the task of casting their spells. A sorcerer or bard needs 8 hours of rest (just like a wizard), after which she spends 15 minutes concentrating. (A bard must sing, recite, or play an instrument of some kind while concentrating.) During this period, the sorcerer or bard readies her mind to cast her daily allotment of spells. Without such a period to refresh herself, the character does not regain the spell slots she used up the day before.
Following the "(just like a wizard)" reference, we learn what rest means:
The wizard does not have to slumber for every minute of the time, but he must refrain from movement, combat, spellcasting, skill use, conversation, or any other fairly demanding physical or mental task during the rest period... If the character does not need to sleep for some reason, he still must have 8 hours of restful calm before preparing any spells.
In other words, the summoner could use up their spells, then stay awake all night, concentrate on being on lookout, and then feel fine the next morning without fatigue or exhaustion. But if they put effort into serving lookout for 8 hours (such as perception skill use to detect threats), then they won't have any of their spells ready.
Regarding the rules for resting on PFSRD:
The PFSRD mentions some non-core rules about accumulating penalties versus sleep effects. It fails to mention that these are optional. These variant rules were mentioned once in a part of Trial of the Beast, which is book 2 of the Carrion Crown adventure path. On page 17, it mentions:
Adventuring at night and fatigue from lack of sleep can heighten the atmosphere of horror. Familiarize yourself with the rules on light (Pathfinder RPG Core Rulebook 172–173) and fatigue and exhaustion (Core Rulebook 567). You might also consider using a variant rule where characters who do not get a full night’s sleep may suffer the effects of fatigue. If a
PC does not get at least 6 hours of sleep, she must make a DC 15 Fortitude save or be fatigued and take a –1 penalty on all other checks and saving throws against sleep effects...
These variant rules were designed as an optional plot device to keep the players moving, because they are at a point in the Carrion Crown story where timing is important. But more importantly, while the synthesist is immune to sleep, these penalties would not affect them anyway. The monster rules that define "immunity" include the following:
Immunities can also apply to afflictions, conditions, spells (based on school, level, or save type), and other effects. A creature that is immune does not suffer from these effects, or any secondary effects that are triggered due to an immune effect.
So if something would cause the synthesist to take a penalty versus sleep effects, the synthesist would ignore it while they have their eidolon's sleep immunity.
