This is a question we are asked all the time, most times the answer is a resounding NO! UNLESS it is for a specific and acceptable purpose, let us go into a few details for you here and discuss types of text.
Headings Text
Colour is acceptable here should you want your headings to stand out more but try to choose a colour that matches your site's overall design. That said, most sites automatically embolden headers and make the text larger, so the colour is not really required to make your headings stand out. Run the book test in your head. Do books or newspapers have coloured headings? No, they don't as a rule the headings are just in larger text and usually bold
General Website Text
This is the text you are reading now, the main text for all the information on your site. This should be a dark colour on a light background. This is the easiest on the eye. Deviate from that and your site will be harder to read, and it could cause some visitors to leave your site early. This could cost you sales or customers.
When Coloured Text Is Acceptable.
Again, we are talking about web standards here. Before you start shouting that you do not want to conform to standards, that is up to you. However, changing what web users are used to adds a level of difficulty to the usability of your site which can result in lost sales and customers.
Acceptable: Red text for warnings created by things like contact forms when a field has been missed or to REALLY emphasise a point the user is agreeing to.
Acceptable: Green text for a confirmation message, EG: Your message has been sent
Acceptable: Blue text or a noticeably different shade from your site's standard text colour for any links. Users expect links to be blue or some other shade of your standard text.
Final Guidelines
DO NOT use multiple colours for text on a web page (Excluding the above standards) it's very unprofessional,
DO stick to black or dark grey text on a white page.
DO use standard, easy to read complimenting fonts.
