What are the specific assumptions of MRS?

I know MRS is independent of prices and compares the ratio of goods which the consumer will exchange one good for the other good. But is the marginal rate of substitution affected by the consumption of those goods or not? It seems trivial but can't get my head around it.

Different consumer preferences will lead to different properties of the consumer's willingness to trade one good for another.

For instance, suppose that consumers preferences are linear in the consumption of either good: $U=\beta_x x + \beta_y y$ where $\beta_x$,$\beta_y$ are positive constants

The marginal utility of $x$ would be $\beta_x$ and the marginal utility of $y$ would be $\beta_y$, and the MRS would be $\frac{\beta_x}{\beta_y}$, also a constant. This tells us the consumer would be willing to trade between $x$ and $y$ at a constant rate.

An alternative would be consumption of each good entering utility in a multiplicative fashion (Cobb Douglas), for instance $U=xy$. The marginal utility of $x$ would be $y$ and the marginal utility of $y$ would be $x$ and so the MRS would be $\frac{y}{x}$, which indeed would depend on consumption. In this example, the MRS decreases as consumption of $x$ rises. As the consumer enjoys more $x$ he is willing to give up less and less $y$ for another unit of $x$.

So it's not entirely trivial whether the MRS depends on consumption, whether it does or not depends on how you model consumer's preferences.

It depends. Hessian gives a good answer, but I think we can say something more.

A utility function $u(x)$ assigns to each bundle of goods $x=(x_1,\ldots,x_n)$ a real number. Along an indifference curve, we have \begin{equation}u(x)=u_0\tag{1}\end{equation} for some real constant $u_0$. Now, the marginal rate of substitution $MRS_{ij}$ of good $i$ for good $j$ exists at a given point $x$ whenever $u_i(x)\neq 0$, for then the implicit function theorem implies that $(1)$ implicitly defines a function $f^i$ such that $x_i=f^i(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n)$, and that we have $$u(x_1,\ldots,x_{i-1},f^i(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n),x_{i+1},\ldots,x_n)=u_0.\tag{2}$$ Differentiating $(2)$ with respect to $x_j$ where $j\neq i$ gives us $$-\frac{\partial f^i}{\partial x_j}\Big|_{u\text{ constant at }u_0}=\frac{u_i(x)}{u_j(x)}=MRS_{ij}.\tag{3}$$ Note that the partial derivatives $u_i(x)$ and $u_j(x)$ are dependent on the bundle of goods $x$, so it is logically possible that $MRS_{ij}$ varies with $x$.

If $MRS_{ij}$ does not vary with $x_j$ and is thus constant with respect to $x_j$ at some level $k$, it follows from $(3)$ by integration with respect to $x_j$ on both sides that $$f^i(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n)=-kx_j+m$$ for some real constant $m$ independent of $x_j$. Thus, in this case, the indifference curve in $(x_i,x_j)$-space is a straight line.