I am trying to figure out the conditions such that $[X, Y]_p$ is linearly independent with $X_p$ and $Y_p$ for some vector fields $X, Y$ and some $p$ in a three-dimensional manifold.
I have that $X_p$ and $Y_p$ are linearly independent, but what else do I need to conclude anything from here?
On
I will state some geometric result : Consider a curve $c$ with $c'(t)= A(t) X_{c(t)} + B(t) Y_{c(t)}$ On ${\bf R}^3$, any two points is connected by $c$ iff $[X,Y],\ X,\ Y$ are independent.
That is if $E_1=(1,0,0),\ E_2=(0,1,0)$, then $(0,0,0),\ (0,0,1)$ is connted by $c$ So the these three vectors are dependent.
You probably won't like this answer, but you need to know that there are no integral manifolds for the $2$-dimensional distribution spanned by $X$ and $Y$. For example, taking $$X=\frac{\partial}{\partial x} \quad\text{and}\quad Y=\frac{\partial}{\partial y}+x\frac{\partial}{\partial z},$$ we have $$[X,Y]=\frac{\partial}{\partial z}.$$ (Of course, if we leave off the $x\,\partial/\partial z$, we have linearly independent, but commuting, vector fields.)
Note that your question was phrased just at one point $p$, but linear independence will be an open condition, so (assuming smoothness, of course) linear independence will hold in a neighborhood of $p$ if it holds at $p$.