Redirecting stdout to a file you don't have write permission on

Yes, using tee. So echo test > /tmp/foo becomes

echo test | sudo tee /tmp/foo

You can also append (>>)

echo test | sudo tee -a /tmp/foo

To replace the content of the file with the output of echo (like the > shell redirection operator).

echo test | sudo dd of=/tmp/foo

To write into the file (at the beginning, though you can use seek to output at different offsets) without truncating (like the 1<> Bourne shell operator):

echo test | sudo dd of=/tmp/foo conv=notrunc

To append to the file (like >>), with GNU dd:

echo test | sudo dd of=/tmp/foo oflag=append conv=notrunc

See also GNU dd's conv=excl to avoid clobbering an existing file (like with set -o noclobber in POSIX shells) and conv=nocreat for the opposite (only update an existing file).

tee is probably the best choice, but depending on your situation something like this may be enough:

sudo sh -c 'echo test > /tmp/foo'