Fish and Visitors stink in 3 days.

Benjamin Franklin (1706-1790) American statesman, scientist, philosopher, aphorist

Poor Richard (1736 ed.)
    (Source)

Another saying Franklin repurposed from other sources. Proverbs comparing things to fish not aging well, or how guests outstay their welcome, or both, are not uncommon over the centuries.

Plautus is often mentioned as the originator of the combined sentiments, but instead he wrote of each individually. In the Asinaria [The Comedy of Asses], Act 1, sc. 3, l. 26 (c. 212-205 BC), he mentions a Roman proverb:

CLEARITA: Quasi piscis itidem est amator lenae: nequam est nisi recens.

[Just like a fish, so is a lover to a procuress; he's good for nothing if he isn't fresh.] [tr. Riley (1912)]
[For a brothel-keeper a lover is like a fish: he's no good unless he's fresh. [Source]
[For a madam a lover is just like a fish: if he’s not fresh, he’s worthless. [Loeb]]

Plautus also wrote in Miles Gloriosus [The Swaggering Soldier], Act 3, sc. 1, l. 146 (206 BC):

Hospes nullus tam in amici hospitium divorti potest, quin, ubi triduum continuum fuerit, jam odiosis siet.

[Whene’er a man is quartered at a friend’s, if he but stay three days, his company they will grow weary of.] [tr. Thomas]
[No guest is so welcome in a friend's house that he will not become a nuisance after three days.] [Source]

Medieval Italy sees the development of the proverb "L'ospite è come il pesce: dopo tre giorni puzza [The guest is like fish: after three days it stinks]." This is said to derive from the Latin "Post tres saepe dies vilescit piscis et hospes," which is sometimes credited (incorrectly) to Plautus. Wegeler includes that Latin in Philosophia Patrum [Philosophy of the Fathers] (1869) as a proverb (No. 931).

Erasmus in his Adagia [Proverbs] (1523), mentioning Plautus and the Asinaria line above, indicates a shortened version of this is still in circulation as a saying to (indirectly) refer to friends who stay three days or more:

Piscis nequam est, nisi recens.

[Fish is bad, unless it's fresh.]

John Lyly wrote in Euphues and His England (1580), "Fish and Guests in three days are stale."

Matthew Henry, in his Bible commentary (1706) on Proverbs 25:17, mentions that Latin proverb "Post tres saepe dies vilescit piscis et hospes" (translating it "After the third day fish and company become distasteful").