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Is the classic Q-learning algorithm, using lookup table (instead of function approximation), equivalent to dynamic programming?

Dynamic Programming is an umbrella encompassing many algorithms. Q-Learning is a specific algorithm. So, no, it is not the same.

Also, if you mean Dynamic Programming as in Value Iteration or Policy Iteration, still not the same. These algorithms are "planning" methods. You have to give them a transition and a reward function and they will iteratively compute a value function and an optimal policy.

Q-Learning is a model-free reinforcement learning method. This one is "model-free", not because it doesn't use a machine learning model or anything like that, but because they don't require, and don't use a model of the environment, also known as MDP, to obtain an optimal policy. You also have "model-based" methods. These, unlike Dynamic Programming methods, are based on learning a model, not simply using one. And, unlike model-free methods, don't throw away samples after estimating values, they instead try to reconstruct the transition and reward function to get better performance.

Model-based methods combine model-free and planning algorithms to get same good results with less amount of samples than required by model-free methods (Q-Learning), and without needing a model like Dynamic Programming methods (Value/Policy Iteration).

From Sutton & Barto's book (Reinforcement Learning: An Introduction, chapter 4)

The term dynamic programming (DP) refers to a collection of algorithms
that can be used to compute optimal policies given a perfect model of
the environment as a Markov decision process (MDP). Classical DP
algorithms are of limited utility in reinforcement learning both
because of their assumption of a perfect model and because of their
great computational expense, but they are still important
theoretically.

So, although both share the same working principles (either using tabular Reinforcement Learning/Dynamic Programming or approximated RL/DP), the key difference between classic DP and classic RL is that the first assume the model is known. This basically means knowing the transition probabilities (which indicates the probability of change from state s to state s' given action a) and the expected immediate reward function.

On the contrary, RL methods only require to have access to a set of samples, either collected online or offline (depending on the algorithm).

Of course, there are hybrid methods that can be place between RL and DP, for example those that learn a model from the samples, and then use that model in the learning process.

NOTE: The term Dynamic Programming, in addition to a set of mathematical optimization techniques related with RL, is also used to refer a "general algorithmic pattern", as pointed in some comment. In both case, the fundaments are the same, but depending on the context may have a different meaning.

If you use Q-learning in an offline setup, like AlphaGo, for example, then it is equivalent to dynamic programming. The difference is that it can also be used in an online setup.