There are a lot of places to find survey questions online. The main two sources are organized, searchable databases (sometimes by topic) and survey documentation websites (where you can view the questionnaires themselves). You can also find them scattered on websites, in publications, and other places. I always advise researchers to remember that they can design their own questions or modify existing ones, and the fact that a question has been used by other researchers doesn't essentially make it a good question (unless your sole purpose is replication).
Here are some of the question repositories and sites I've found most helpful (or recommended by others). Databases and static documentation sites are intermingled below, and sorted by topic when possible.
Descriptions to come as I have time and a chance to use them. Feel free to suggest additions.
General Demographic, Social, and Omnibus Survey, and Poll, Questions
American Community Survey Questionnaire Archive (U.S. Census Bureau)
European Social Survey (ESS) Questionnaire
ICPSR Web Site
We usually think of ICPSR as a data warehouse, but I've found that when you search their database you get question text with summaries of responses. So it serves dual purpose.
A product of NCHS. Holds cognitive interview and pretesting results as well as question text.
Survey Monkey's International Question Bank | SurveyMonkey Blog
Health Survey Questions
Behavioral Risk Factor Surveillance System (BRFSS) State-Added Question Database
This is the best place to find state-added BRFSS questions by state and topic
Behavioral Risk Factor Surveillance System (BRFSS) Questionnaires (CDC)
California Health Interview Survey (CHIS) Questionnaires
Cancer Questions from Grid-enabled Measures (GEM) Database
National Health Interview Survey (NHIS) - Questionnaires
NQF: Quality Positioning System
Used by hospitals for quality of care questions
Patient Reported Outcome Measurement System (PROMIS) Database
SHADAC's State Reform Survey Item Matrix (SRSIM)
A well-designed Excel spreadsheet with question text from various states
Religion Survey Questions
Congregational Survey Question Bank
Sexuality Survey Questions
Questionnaire for Gay Sentiment Study
Sexual orientation questions on LGBTData.com
The most comprehensive resource for LGBT questions and data sources I've found yet.