Given an IMDB movie id, how to programmatically get its poster image

You can use IMDB API located at

www.omdbapi.com

You can search for movie names like this -

http://www.omdbapi.com/?t=hangover%202

{
"Title":"The Hangover Part II",
"Year":"2011",
"Rated":"R",
"Released":"26 May 2011",
"Genre":"Comedy",
"Director":"Todd Phillips",
"Writer":"Craig Mazin, Scot Armstrong",
"Actors":"Bradley Cooper, Zach Galifianakis, Ed Helms, Justin Bartha",
"Plot":"Right after the bachelor party in Las Vegas, Phil, Stu, Alan, and Doug jet to Thailand for Stu's wedding. Stu's plan for a subdued pre-wedding brunch, however, goes seriously awry.",
"Poster":"http://ia.media-imdb.com/images/M/MV5BMTM2MTM4MzY2OV5BMl5BanBnXkFtZTcwNjQ3NzI4NA@@._V1_SX320.jpg","
Runtime":"1 hr 42 mins",
"Rating":"7.1",
"Votes":"13547",
"ID":"tt1411697",
"Response":"True"
} 

You can get the "ID" (tt1411697) from this JSON response.

IMDB blocks the direct linking of images from their site on other sites, I think this previous question covers the topic.

The easiest way to get around this is to download the image and host it yourself rather than linking IMDB's copy. Alternatively you could investigate alternative movie DBs to see if they can offer what you want - the answers to this question on IMDB APIs lists a few. The Movie DB API looks like a good bet.