Question: Let $G$ be an ample $\mathbb{Q}$-divisor on a smooth variety $X$. Let $D$ be a $\mathbb{Q}$-divisor linearly equivalent to $G$. Let $f: Y\to X$ be a common log resolution of $G$ and $D$. We define the multiplier ideal of a divisor $G$ as $I(G)=f\_\*O\_Y(K\_{Y/X}-[f^\*G])$. Are the multiplier ideals $I(G)$ and $I((1-t)G+tD)$ the same for sufficiently small $t>0$?

Answer: 
I don't think so. Take for example $X=\mathbb{P}^2$, and $G$ and $D$ to be distinct lines.
then $I(G)=\mathcal{O}\_X(-G)$ while $I((1-t)g+tD)=\mathcal{O}\_X$ for every small $t>0$.
Maybe you might want to look at the multiplier ideal associated to the linear series $|G|$.
Think about it this way: The co-support of the multiplier ideal of $G$ is the locus where the pair $(X,G)$ has worse than klt singularities. If $(X,G)$ has non-klt singularities, say because some of its coefficients are at least $1$, and $(X,D)$ is dlt, then some of the small perturbations will likely be still klt, so you get a jump in the multiplier ideal. A simple concrete example for this is given by Gianni Bello in his answer.