They aren't necessarily inaccurate but you have the highest chance of accuracy if you use first morning urine.
Any time of day can be accurate if your urine is concentrated enough. This means that you haven't urinated in several hours, but have not drank a lot either.
If your urine is too diluted then it will contain less of the hormone HCG and be harder to detect.
This really only makes a difference in very early pregnancy. Later in the first trimester there should be enough of the hormone to be picked up regardless.
Most tests these days are accurate and can be taken 4 days before your missed period. Pink dye test (like First Response) are the most accurate as they are more sensitive than the blue dye ones. I still always recommend taking the test the day or after your period is due because it will be more accurate, and if pregnant, then the hormone levels the test is looking for is bound to be highly (the double every 24-48 hours in pregnancy). Tests can be taken 2-3 weeks after intercourse if you're not sure about when your period is due.
Most pregnancy tests now a days can detect pregnancy early before your missed period. But it also depends on the person. I took my pregnancy test later in the day and I was about 4 weeks along when I found out with my daughter and it came out positive. If you are far along for the HCG to be detected through a HPT then it doesn't matter what time of day you take it