Explanation for regression coefficient $\beta= 0$ and standard error $\sigma(\beta) = 0$ one of the coefficients in an OLS regression turned out zero and its Standard error is zero as well. Would you be suspicious of this result? Is there any possible explanation for this?
 A: It's very likely it was caused by perfect fit.
It's not wrong actually, see How to derive the standard error of linear regression coefficient.
The standard errors depend on the residual sum of squares (RSS): if it's zero they tend to zero as well.
Other packages/software may give you approximately zero standard errors, but analytically they should be exactly zero.
A: if a $\beta_i=0$ and its $\sigma(\beta_i)=0$, it means the linear regression model wasn't able to find a linear relationship between the dependent variable $y$ and independent variable $x_i$.
This doesn't mean there is no relationship between $y$ and $x_i$. There could instead be a non-linear, or other, interaction going on between them since linear regression will only be able to model linear problems.
The inclusion of other covariates $x_{\neg i}$ within the multiple regression formula might also have an effect on the coefficient estimate assigned to $x_i$, so try stage-wise regression by gradually including or omitting variables and see how parameters might change.
