The rain is "lifting"

I don't think the OED citations apply. I think lift in this case is probably more to do with rain's intensity described as "heavy" and "light" there for "the rain is lifting" means becoming less severe. The support for the above references is that, in my opinion rarely used, when the rain is completely gone one can say "the rain has lifted."

According to the OED to lift referring to rain means to cease temporarily in AmE probably by analogy with other weather phenomena (fog, etc.)

To lift:

• Of clouds, fog, etc.: To rise and disperse. Also (U.S.) of rain: To cease temporarily.

Usage examples:

• 1834 M. Scott Cruise Midge vi. (1842) 102 The clouds..lifted from the eastern horizon majestically slow.

• 1858 Froude Hist. Eng. III. 349 One morning when the darkness lifted, sixty strange sail were found at anchor in the Downs.

• 1870 E. Peacock Ralf Skirl. II. 178 The thick fog had lifted. 1901

• fig. 1897 M. Kingsley W. Africa 232 My..head-ache..soon lifted. -

According to the definition of lift in Cambridge Dictionary it actually makes sense.

Lift

To go away until none is left.

This meaning of 'lift', the range of meaning that includes "a rain lifting", is part of a much larger montage or (if only we knew enough to see the connectedness of the whole) tapestry of meaning that might be said to explain any one use. The range of meaning that includes rain lifting also includes all atmospheric phenomena that increase visibility as they cease: darkness itself lifts, as does snow, hail, fog, smog, etc. So also are curtains and drapes 'lifted' when in fact they may only be moved aside.

The continued application of 'lift' to describe the cessation of atmospheric phenomena that increase visibility as they decrease testifies to the primacy of appearance as a driver of meaning. So, we say "the rain lifted" for much the same reason the 'solstice' (from "sōl sun ... + participial stem of sistĕre to stand still", OED Online) is called the solstice: on that day, to an observer from the limited perspective of earth, the sun appears to stand still. The rain that 'lifts' does appear to vanish into the gradually revealed sky...and an Old Norse word meaning 'sky' ("Old Norse lopt air, sky = lift n.¹" op. cit.) is a source of the first attested uses of 'lift' in English. In those uses the original meaning of 'lift', that is, 'the air, atmosphere, sky', a sense now mostly obsolete, is preserved:

lift, n.¹
....
Obs. exc. Sc. and poet.
The sky, upper regions; †in early use also, the air, atmosphere. Also pl., the (seven) heavens.
OE Beowulf 2832 Nalles æfter lyfte lacende hwearf.
c1000 Sax. Leechd. II. 146 Romane him..worhton eorþ hus for þære lyfte wylme & æternesse.
OE Maxims II 39 Fugel uppe sceal lacan on lyfte.
c1175 Lamb. Hom. 79 Of þe uisces iþe wetere and fuȝeles iþe lufte.

["lift, n.1". OED Online. December 2015. Oxford University Press. http://www.oed.com/view/Entry/108135 (accessed January 03, 2016).]

The foregoing explanation, while complete in itself, discounts the virga effect by relegating it to a secondary role in the continued use of 'lift' to describe the lifting of rain. This is because, while the virga effect applies to rain, it does not apply equally to darkness lifting, or the lifting of hail, sleet, or other atmospheric phenomena. The virga effect does, however, apply uniquely to rain, and may partly explain the continued use of 'lift' to describe the cessation of rain.

Simply, the virga effect, and 'virga' itself, names this phenomenon:

virga, n.
....
2. Meteorol. Streaks of precipitation that appear to be attached to the undersurface of a cloud and usually evaporate before reaching the ground.

Raindrops, as it happens, start large in the clouds but, due to evaporation, diminish as they fall, and especially so in dry air. The raindrops also disperse as they fall, and so many rain showers appear lighter at the lower levels than the upper levels when viewed from a distance. In very dry air, it is common to be able to see in the distance a rain shower that vanishes altogether before reaching the ground. As such a shower gradually shrinks and ceases, then, the rain appears to lift into the cloud that produced it, disappearing from the bottom up, rather than the top down.