Solution:

Label the squares using coordinates, letting the top left corner be $(0,0)$. The burrito will end up in $10$ (not necessarily different) squares. Call them $p_{1} = (x_{1}, y_{1}) = (0,0), p_{2} = (x_{2}, y_{2}), \ldots, p_{10} = (x_{10}, y_{10})$. $p_{2}$ through $p_{10}$ are uniformly distributed throughout the square. Let $d_{i} = |x_{i+1} - x_{i}| + |y_{i+1} - y_{i}|$, the taxicab distance between $p_{i}$ and $p_{i+1}$.

After 1 minute, the pigeon will eat $10\%$ of the burrito. Note that if, after eating the burrito, the pigeon throws it to a square taxicab distance $d$ from the square it's currently in, it will take exactly $d$ minutes for it to reach that square, regardless of the path it takes, and another minute for it to eat $10\%$ of the burrito.

Hence, the expected number of minutes it takes for the pigeon to eat the whole burrito is
$$\begin{aligned} 1 + E\left(\sum_{i=1}^{9} (d_{i} + 1)\right) & = 1 + E\left(\sum_{i=1}^{9} 1 + |x_{i+1} - x_{i}| + |y_{i+1} - y_{i}|\right) \\ & = 10 + 2 \cdot E\left(\sum_{i=1}^{9} |x_{i+1} - x_{i}|\right) \\ & = 10 + 2 \cdot \left(E(|x_{2}|) + E\left(\sum_{i=2}^{9} |x_{i+1} - x_{i}|\right)\right) \\ & = 10 + 2 \cdot \left(E(|x_{2}|) + 8 \cdot E(|x_{i+1} - x_{i}|)\right) \\ & = 10 + 2 \cdot \left(4.5 + 8 \cdot \frac{1}{100} \cdot \sum_{k=1}^{9} k(20 - 2k)\right) \\ & = 10 + 2 \cdot (4.5 + 8 \cdot 3.3) \\ & = 71.8 \end{aligned}$$