This has nothing to do with amsart, but it's how LaTeX has to specify colors.
If you do \color{blue} inside a group, LaTeX will issue a suitable \color instruction after the group ends in order to restore the color that was in force before the change. This is done via a “whatsit”, because TeX has no color support by itself.
In your case, this whatsit is part of the text after the math display and thus will produce an empty line before the end-of-paragraph command (the empty line) is scanned.
Solution: use \begingroup...\endgroup so the whatsit will be placed inside the display. Or, equivalently, \mathcolor.
%\documentclass{amsart}
\documentclass{article}
\usepackage{amsmath}
\usepackage{xcolor}
\begin{document}
Some text.
\begin{equation}
\text{A formula: }a=b+c.
\end{equation}
Some more text, starting new paragraph.
\begin{equation}
\mathcolor{blue}{\text{Another formula: }d=e-f.}
\end{equation}
Some more text, starting new paragraph.
Some more text, starting new paragraph.
\begin{equation}
\color{blue}\text{Another formula: }d=e-f.
\end{equation}
Some more text, not starting new paragraph.
Some more text, starting new paragraph.
\begin{equation}
\begingroup\color{blue}\text{Another formula: }d=e-f.\endgroup
\end{equation}
Some more text, starting new paragraph.
\end{document}
The example shows various alternatives.

Another possibility is to not leave a blank line, and use \indent in front of the text.
Such cases should be rare: how many times do you need to start a new paragraph after a display?