To determine whether the line $EF$ can be tangent to the $A$-excircle of $\triangle ABC$, where $\angle ABC$ is obtuse, we start by analyzing the geometric properties involved.

### Step 1: Understanding the Geometry
1. Excircle Properties: The $A$-excircle is a circle in the exterior of $\triangle ABC$ that is tangent to $BC$ and the extensions of $AC$ and $AB$. It is centered at the excenter $I_A$ opposite to vertex $A$.

2. Altitude Feet: Points $E$ and $F$ are the feet of the perpendiculars from $B$ and $C$ to the lines $AC$ and $AB$, respectively.

### Step 2: Analyzing Line $EF$
- The line $EF$ is the line segment connecting $E$ and $F$, which lie on the sides $AC$ and $AB$ or their extensions due to the obtuseness of $\angle ABC$.

### Step 3: Tangency Condition
- For line $EF$ to be tangent to the $A$-excircle, it would require that the perpendicular distance from the excenter $I_A$ to line $EF$ be equal to the radius of the $A$-excircle.

### Step 4: Geometric Constraints
- Since $\angle ABC$ is obtuse, the altitude $BE$ will lie inside the triangle, and $CF$ will lie outside the triangle.
- This configuration suggests that line $EF$ intersects or passes quite far from the $A$-excircle in relation to the sides $AC$ and $AB$.
- The specific positions of $E, F$ in relation to the $A$-excircle imply that $EF$, instead of being tangent, intersects with circles tangent or externally relative to the triangle sides.

### Conclusion
Given the constructions and the restrictions associated with an obtuse angle and the nature of excircles, $EF$ cannot simultaneously touch the $A$-excircle. Therefore, line $EF$ cannot be tangent to the $A$-excircle.

$$\boxed{\text{Line } EF \text{ cannot be tangent to the } A\text{-excircle.}}$$

The reason derives from the incompatibility of the areas of tangency of the $A$-excircle with the line segment $EF$ as positioned relative to an obtuse $\angle ABC$.