What notable automaton models have polynomially-decidable containment? I'm trying to solve a particular problem, and I thought I might be able to solve it using automata theory. I'm wondering, what models of automata have containment decidable in polynomial time? i.e. if you have machines $M_1, M_2$ you can test if $L(M_1) \subseteq L(M_2)$ efficiently.

The obvious ones that come to mind are DFAs and reversal-bounded counter machines where the number of counters is fixed (see this paper).

What other notable classes can be added to this list?

The more powerful the automata, the better. For example, DFA's aren't enough to solve my problem, and the counter machines can't do it with a fixed number of counters. (Naturally, if you get too powerful, then containment is either intractible, like for NFA's, or undecidable, for CFG's).
Visibly pushdown automata (or nested word automata, if you prefer working with nested words instead of finite words) extend the expressive power of deterministic finite automata: the class of regular languages is strictly contained within the class of visibly pushdown languages. For deterministic visibly pushdown automata, the language inclusion problem can be solved in polynomial time. For more details, see the paper by Alur and Madhusudan, especially Chapter 6.

By the way, the nondeterministic variant of visibly pushdown automata is exponentially more succinct than the deterministic variant, but there the language inclusion problem is EXPTIME-complete and thus intractable. 

Alur, R.; Madhusudan, P. (2009). "Adding nesting structure to words". Journal of the ACM 56(3): 1–43.