Variation of cone angle with concentration of fuel-air mixture
The cone angle of the Bunsen flame is determined by the flame speed of the mixture and the average flow velocity. To first approximation one can write: $$ \sin \alpha = \frac{S}{U} $$
where $\alpha$ is the half cone angle, $S$ is flame speed and $U$ is flow velocity.

When the mixture is made fuel-lean, depending on how flame speed changes the cone angle will change accordingly.

Usually flame speed for fuel-lean mixture is less than that for stoichiometric mixture. Therefore we can expect the cone angle to decrease.