(Essentially reposting Jesus Martinez Garcia and Karl Schwede's comments as an answer)
If you are blowing up a torus-invariant sheaf of ideals, then the Rees algebra (the thing you take Proj of to get the blow-up) has grading by characters of the torus, so the blowup has an action of the torus, so it is toric (since it also contains a dense copy of the torus that acts on it). Thus, if your weighted blow-up is weighted "along the coordinate hyperplanes", it will be toric.
If you blow up a non-invariant sheaf of ideals, you do not get something toric in general. If you blow up a non-invariant point of $\mathbb A^2$, then the blow-up map already cannot be toric, even though both varieties are toric. Blowing up two different points of $\mathbb A^2$ gives a total space which has no toric structure at all: the two (-1)-curves must be torus-invariant and blowing them down gives a toric map to $\mathbb A^2$, but any toric blow-up of $\mathbb A^2$ must have the exceptional locus lying over a single point.