Pulled this up from NODE; Multiple origins seem to be the case
LIME1
noun (also quicklime) [mass noun] a white caustic alkaline substance consisting of calcium oxide, which is obtained by heating limestone and combines with water with the production of much heat.n
(also slaked lime) a white alkaline substance consisting of calcium hydroxide, made by adding water to quicklime.
n(in general use) any of a number of calcium compounds, especially calcium hydroxide, used as an additive to soil or water.
n archaic birdlime.

verb [with obj.]
1 treat (soil or water) with lime to reduce acidity and improve fertility or oxygen levels.n [often as adj.] (limed) give (wood) a bleached appearance by treating it with lime: limed oak dining furniture.

2 archaic catch (a bird) with birdlime.
DERIVATIVES
limy adjective (limier, limiest).
ORIGIN Old English līm, of Germanic origin; related to Dutch lijm, German Leim, also to loam.

LIME2 noun
1 a rounded citrus fruit similar to a lemon but greener, smaller, and with a distinctive acid flavour.n [mass noun] a drink made from or flavoured with lime juice: lager and lime.

2 (also lime tree) the evergreen citrus tree which produces this fruit, widely cultivated in warm climates.
• Citrus aurantifolia, family Rutaceae.

3 [mass noun] a bright light green colour like that of a lime: [as modifier] dayglo orange, pink, or lime green.
ORIGIN mid 17th cent.: from French, from modern Provençal limo, Spanish lima, from Arabic līma; compare with lemon.

LIME3 (also lime tree)
noun a deciduous tree with heart-shaped leaves and fragrant yellowish blossom, native to north temperate regions. The pale timber is used for carving and cheap furniture. Also called linden.
• Genus Tilia, family Tiliaceae: many species, including the widely grown hybrid common lime (T. × europaea), and the small-leaved lime (T. cordata), which dominated the pre-Neolithic forests of much of lowland England.
ORIGIN early 17th cent.: alteration of obsolete line, from Old English lind (see linden).

LIME4 W. Indian
verb [no obj., with adverbial] sit or stand around talking with others: boys and girls were liming along the roadside as if they didn't have anything to do.

noun an informal social gathering characterized by semi-ritualized talking.
ORIGIN origin uncertain; said to be from Limey (because of the number of British sailors present during the Second World War), or from suck a lime, expressing bitterness at not being invited to a gathering.