Is it necessary to obtain permissions for copying figures from published articles in your proposal/dissertation?
For figures from your own papers, it would depend on the copyright transfer agreement you (or the corresponding author) signed upon publication. However, all copyright agreements I know explicitly authorize reuse of content for academic theses.

As an example: the American Chemical Society, which does not leave the authors too many rights, includes this wording:


  Authors may reuse all or part of the Submitted, Accepted or Published Work in a thesis or dissertation that the Author writes and is required to submit to satisfy the criteria of degree-granting institutions. Such reuse is permitted subject to the ACS’ “Ethical Guidelines to Publication of Chemical Research"




For figures from others’ papers, a thesis is not very different from any other publication (see the related question about blogging). Unless your institution has a specific agreement with publishers (as MIT seems to have), you have to either:


ask for permission; these days, it's all done online and once you have located the appropriate form for a publisher, you can make your requests and get all the answers the next day
rely on fair use in the US, or similar law in other countries; around me, most people actually do that, either knowingly or just out of ignorance :)