No, but they do, typically, operate at lower speeds. In really, really bad storms (like on 8/8/2007), the subway can shut down. On 8/8/07, 3 inches of rain fell in one hour, and a tornado hit Brooklyn, causing most subway lines to be severely delayed or completely shut down.

Rarely, yes. It takes a pretty serious storm, but I've seen service interrupted due to rain within this past year. This year has had record rainfall.

http://www.nytimes.com/2009/08/01/nyregion/01hot.html

"One reason for the record-breaking low temperatures was the record-breaking rainfall and accompanying cloud cover. The deluge of more than 16 inches (twice the average amount) made it the second wettest June and July on record, edging 1928. The wettest June and July was 1975, with more than 19 inches."

(First reference I could Google up.)