Missing final brackets (in one equation { and in other - ]) in the equations

You can't use \left in one line of a split and \right in another one. You should also use an align* environment (and don't use redundant \left and \right); the big delimiters must be set by hand.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
F'&=\delta m\biggl\{f\frac{M_1l}{r^3}\bigl[3\cos^2\phi\cos^2(\lambda-D)-1\bigr]
    +f\frac{M_2l}{R^3_1}(3\cos^2\phi\cos^2\lambda-1)-{} \\
  &\qquad{}-\frac{1}{2}f\frac{M_2r_1}{R^3_1}\cos\phi\cos(\lambda-D)(1+3\cos2D)
   -\frac{3}{2}f\frac{M_2r_1}{R^3_1}\cos\phi\sin(\lambda-D)\sin2D\biggr\} \\[2ex]
F^n&=\delta m\biggl\{-\frac{3}{2}f\frac{M_1l}{r^3}\sin 2\phi\cos^2(\lambda-D)
    -\frac{3}{2}f\frac{M_2l}{R^3_1}\sin 2\phi\cos^2\lambda-{} \\
   &\qquad{}-\frac{1}{2}f\frac{M_2r_1}{R^3_1}\sin\phi\bigl[3\sin(\lambda-D)\sin2D-
    \cos(\lambda-D)(1_3\cos 2D)\bigl]\biggl\} \\[2ex]
F^m&=\sigma m\biggl[-\frac{3}{2}f\frac{M_1l}{r^3}\cos\phi\sin 2(\lambda-D)
    -\frac{3}{2}f\frac{M_2l}{R^3_1}\cos\phi\sin 2\lambda+{} \\
   &\qquad{}+\frac{1}{2}f\frac{M_2}{R^3_1}r_1\sin(\lambda-D)(1+3\cos 2D)
    +\frac{3}{2}f\frac{M_2}{R^3_1}r_1\cos(\lambda-D)\sin 2D\biggr]
\end{align*}
\end{document}

• First of all I did an align*, so the three equals sign can be aligned to each other. It's not good to stack \[...\] formulas.

• I removed all the inner \left and \right that do nothing except adding unwanted spaces. However, I increased the size of a [...] pair to make clearer their correspondence (it's in the second equation; there's no need for doing this in the first formula)

• In order to make clear that each second line is a continuation, I added a \qquad of space to push it to the right of the alignment point.

• Before or after the "isolated" minus or plus signs, I put {} in order to get correct spacing, otherwise they would not work as binary operation because of how TeX determines the difference between $-1$ and $2-1$.

• Most important, I set by hand the size of the main delimiters, because so you have full control over them even if they are in different lines.

As a side note, I wouldn't repeat the operation sign at the break point; it's a bad habit of Russian typography, that's not used much in Western countries. I find it distracting and ambiguous: in the first equation is it "minus minus" that makes "plus"? It isn't, I know, but why repeating it? The reader finds the break, goes on the next line where it's clear that the formula continues.