Warning: Some links are broken. This is an area I have not maintained since I retired. The links might be fixed/removed at some future date (posted Nov. 2020).
1) One or two AP Chemistry Free Response questions and solutions
You can obtain more by searching for specific years. More recent material (back to 1999 or so) comes from the Collegeboard. Material earlier than that is due to various individuals not associated with the Collegeboard posting questions and answers.
These tests are used in the early selection phases for the USA team which is sent to the International Chemistry Olympiad. The two test types are called the "Local Exam" and the "National Exam."
3) Australian Chemistry Olympiad
These tests are used in the early selection phases for the Australian team sent to the IChO. The two test types are called the "National Qualifying Exam" and the "Final Qualifying Exam."
4) Questions from past IChO Tests
The links on this site go to IChO tests (both theoretical and lab practical tests) as well as test prep questions prepared by each IChO host. I recommend you visit each host site as well. There may be additional files of interest. Also, some of the links are misnamed. In one case, a link opens problem 4 while the text indicates problem 3. The link with the problem 4 test opens problem 3. In another case, the test answers are opened by both links - one for questions and one for answers. To get the question file, open the subdirectory index and download the appropriate file. Another site of possible IChO interest is here.
5) Questions from past New York Regents tests
You will find a menu of many past New York Regents tests. Chemistry is in the science section (scroll down to the bottompart of the menu) and the earliest test dates back to August 1974.
6) Higher School Certificate exams; New South Wales, Australia.
Each year has an alphabetical list of exams, so you will need to scroll down a bit for chemistry. These tests extend back to 1995.
7) The Ohio State University: Quizzes and Tutorials.
The test questions are divided up into 15 topics, with a large number of three-question quizzes under each of the 15 topics. These questions are designed to be answered on-line. Each quiz has a link to a tutorial about the material being tested.
ConcepTests are intended for use in large-group instruction and not so much for use in a graded test. If you find the concept interesting, try searching on "ConcepTest."
9) Bank of Chemistry Questions and Problems
Brother Malcolm Melcher has almost 7,000 test questions and problems, assembled over 39 years of chemistry teaching. They are all in a password-protected test bank and you have to contact him for a password. Students, this is a teachers-only thing. Sorry.
10) Released Questions from the Massachusetts Comprehensive Assessment System
You need to select each year in turn and scroll down near the bottom for chemistry. The files names for different years' tests are, in some cases, the same, so you may have to do some renaming of files. Each chemistry year also includes examples of student work for several questions.
11) Released Questions from the California Standards Test
Scroll down to near the bottom for chemistry.