If the loss tangent is too high then you will lose more power in the antenna. It can make the board hot. 0.001 is low loss tangent, 0.1 is bad. In between you need to work out whether it matters to you. Sometimes it is cheaper to waste power (that you have to pay for in bigger batteries and higher spec amplifiers) making the board warm or hot than to buy a more expensive board.
FR4 is cheap but may not be good enough for efficient use above 5 GHz (or even below 5 GHz).
When changing the dielectric constant, impedance mismatch of the antenna occurs, and radiation efficiency quickly degrades. The radiation efficiency is given by:
Efficiency= (𝑃out) / (𝑃in-𝑃mismatch) (1)
𝑃out=𝑃in-𝑃mismatch-𝑃diel-𝑃tissue (2)
where 𝑃in is the input power of the antenna,𝑃mismatch is the power loss due to antenna-source mismatch, 𝑃diel is the dielectric loss, 𝑃tissue is the loss absorbed by the tissue material, which is dominant, and 𝑃out is the total power received at the outside of the tissue block. From equation (2), the 𝑃out is decrease when the dielectric loss increased. From equation (2), the antenna efficiency is decrease, so we prefer to choose substrate material have a less tangent.