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I have a requirement to design minimax filter with linear programing (linprog in MATLAB). To build the filter I must choose vector $\omega$ (frequency sample), how many values I need to take to get the optimal result? How to spread them across the interval $[0,\pi]$?

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There is no real rule, but you would usually choose around $10N$ frequency points, where $N$ is the desired filter order. Distribute the points equidistantly over pass band(s) and stop band(s). This does not mean that they are equidistant in the interval $[0,\pi]$ because usually there is at least one "don't care" region where no desired response is specified (such as in the transition band between pass band and stop band).

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