Does anyone recognize this inequality?

Use Abel transform: denote $x_i=y_1+y_2+\dots+y_i$, then $$\sum \alpha_i x_i=\sum y_i (\alpha_i+\alpha_{i+1}+\dots+\alpha_n). $$ We have $\alpha_i+\alpha_{i+1}+\dots+\alpha_n=N-(\alpha_1+\dots+\alpha_{i-1})\geqslant N-i+1$ for $i=1,\dots,N$. Therefore $$ \sum \alpha_i x_i\geqslant Ny_1+(N-1)y_2+\dots+y_N=x_1+\dots+x_N. $$

$\newcommand{\al}{\alpha}$ The following, rather intuitive proof is done by induction on $n$. The case $n=1$ is trivial. Suppose now that $n\ge2$. By continuity, without loss of generality $x_1<\dots<x_n$. The minimum of $\sum_1^n\al_i x_i$ over all $\al=(\al_1,\dots,\al_n)$ as in the OP is attained. Let $\al=(\al_1,\dots,\al_n)$ be a point of such an attainment.

To obtain a contradiction, suppose that $\al_1<1$. Then the condition $\sum_1^n\al_i=N\ge1$ implies that $\al_j>0$ for some $j\in\{2,\dots,n\}$. Replacing $\al_1$ and $\al_j$ respectively by $\al_1+h$ and $\al_j-h$ for a small enough $h>0$, we get a smaller value of $\sum_1^n\al_i x_i$ (because $x_1<x_j$). This contradicts the assumption that $\al$ is a point of minimum of $\sum_1^n\al_i x_i$.

So, $\al_1=1$, and your inequality reduces to $\sum_2^n\al_i x_i\ge\sum_2^N x_i$ given that $\al_i\in[0,1]$ for all $i$ and $\sum_2^n\al_i=N-1$, and the latter inequality is true by induction.

(In a way this is a rephrasing of Iosif's answer, but from a different perspective. My main intention is to point out the connection to matroids.)

The inequality follows from:

Fact. The polytope $K = \Bigl\{\alpha \in [0,1]^n: \sum_{i = 1}^n\alpha_i = N\Bigr\}$ is the convex hull of indicator vectors of subsets of $[n] = \{1, \ldots, n\}$ of size $N$.

Assuming this fact, and because any linear function over a polytope achieves its minimum at a vertex, we have that $\sum_{i = 1}^n{\alpha_i x_i} \ge \sum_{i \in S} x_i$ for some set $S$ of size $N$. The right hand side is then clearly minimized for $S = \{1, \ldots, N\}$ if $x_1 \le x_2 \le \ldots \le x_n$.

To show the Fact, we need to show that every extreme point $\alpha$ of $K$ is an indicator vector of a set of size $N$. This is clearly true if $\alpha \in \{0,1\}^n$, so assume, towards contradiction, that for some $i$ we have $0 < \alpha_i < 1$. Then there must be at least one other $j \neq i$ such that $0 < \alpha_j < 1$, and we can write $\alpha$ as a convex combination of two vectors in $K$, one in which we add a tiny $h$ to $\alpha_i$ and subtract $h$ from $\alpha_j$, and one in which we reverse the signs. This means that $\alpha$ is not an extreme point, a contradiction. You can see that this argument is essentially the same as Iosif's.

Another way to state the Fact is that $K$ is the base polytope of the uniform matroid over $[n]$ of rank $N$. A vast generalization is the characterization of the facets of the base polytope of any matroid, due to Edmonds: see, e.g., Section 10.7 in these notes.