What does coefficient mean for a binary independent variable in multiple linear regression?
Your question is a little bit confusing but I will try to answer it:

You are asking if the coefficient sign equals the correlation sign? You want to know if a positive sign in the regression leads to a positive correlation between the $Y$ and the binary $X$s.

Depends.

This is true if you are running a regression with only one THAT regressor.

The equation $Retweets=\beta_0+\beta_1*truth.label$ complies with this.

The model $Retweets=\beta_0+\beta_1*truth.label.1+\beta_2*truth.label.2$ does not necesarily complies with the statement.

Why? When you have two or more variables in a model, what they do to each other is to complement between themselves, i.e. one finds what the other variable misses (in terms of errors). So you can't state anything about coefficients and correlations.

This is true, does not matter the type of variable: Binary, Integer, Continuos.