In his edition of Romeo and Juliet for the New Cambridge Shakespeare (Cambridge University Press, 1984), G. Blakemore Evans points out that the idea express in Friar Lawrence's words ("The sweetest honey / Is loathsome in his own deliciousness" ) was proverbial in Shakespeare's time. This is something the two existing answers have overlooked.
The traditional reference for this sort of proverbial wisdom is M. P. Tilley's A Dictionary of the Proverbs in England in the Sixteenth and Seventeenth Centuries (1950). The entries in Tilley's dictionary are numbered; the relevant entry is H560: "It is not good to eat too much honey". The ultimate source given there is Proverbs 25.27 (King James Version):
It is not good to eat much honey: (…)
The idea behind Friar Lawrence's words can also be found in A Midsummer Night's Dream Act 2, scene 2, where Lysander says,
For, as a surfeit of the sweetest things
The deepest loathing to the stomach brings,
The idea that too much of a sweet thing (in the sense of "pleasant") can cause loathing can even be transferred to music, as in Duke Orsino's words at the beginning of the play Twelfth Night:
If music be the food of love, play on,
Give me excess of it, that, surfeiting,
The appetite may sicken, and so die.
Duke Orsino is love sick, and "appetite" here refers to his longing for music. A few lines down, after listening to more of the music, Orsino says,
Enough, no more!
'Tis not so sweet now as it was before.
 The second quarto of 1599, which G. Blakemore Evans used as the basis for the New Cambridge Shakespeare edition, has "his own deliciousness". Of course, the meaning is "its own deliciousness". Some other editors therefore replace "his" with "its".