"If hard determinism is true, and our thoughts are merely the results of a causal chain of atomic interactions, are reason and logic illusory?"
In the sense Lennox is talking about, no. Reason and logic can arise deterministically if you set up a system that deterministically evolves by pruning out anything irrational and illogical.
Lennox relies on the strawman argument that because evolution by natural selection involves a "random" mechanism, not guided by any intelligence, that the result is necessarily random and unintelligent. If the rules and principles by which my reasoning worked were selected randomly, from all possible rules, it would be highly unlikely to generate reliable conclusions. It's a variant on the Argument from Design.
But evolution by natural selection is not random. It consists of two parts: random variation, and highly non-random selection. It is the selection that contributes the 'design'. A good analogy for it is topiary, the shaping of plants and bushes into artificial sculpture-like shapes. The plant grows randomly. But the gardener cuts off any branch that crosses a chosen line in space. The result is a plant that grows in a highly non-random form that precisely avoids crossing the lines. How, you might ask, could such a shape so perfectly suited for survival arise by the random growth of twigs and branches? How did the plant know so precisely when and where to stop growing? That's the wrong question! The source of the 'design' in natural evolution is not random mutation, but in the highly non-random process of survival of the fittest.
We can trust our evolved logic and reason to give us reasonably reliable answers about the world because those alternatives that gave unreliable answers all died out long ago. Only accurate logic and reason enables us to survive. Our minds are shaped by the dangers in our environment. And it is only by such a process that the genesis of reason and logic can be explained. Intelligent design simply shifts the question back a step, you have the same problem to explain the designer.
Besides the question Lennox raised, which was more about physics being unguided than it being deterministic (there is no such thing as 'randomness' in a deterministic universe!), there is a separate question whether hard determinism poses any obstacle to human reason and logic. Here we run into questions of 'free will' and the meaning of 'counterfactuals', on which human reason in aid of decisionmaking is often supposed to rely, but which are commonly held to be incompatible with determinism and materialism. That's a far more complicated debate.
Human reasoning builds simplified predictive models of the world. It divides it up into 'objects' which have their own models. A model takes a set of initial facts about the world as input, some known, some unknown, and some of which are under our control, and generates a predicted outcome as output. The problem-solving mind can then run through all the possible inputs to find which ones are most likely to give a desirable output, and do that. We model counterfactuals - things that could happen but don't - as an inherent and essential part of our reasoning process. But in a deterministic universe, the idea of "something that could happen but doesn't" is a nonsense! There is only one thing that could and does happen.
We model other humans and animals recursively, as "agents". This treats minds as black boxes that make decisions autonomously, and we use our own mind, with a modified set of inputs to account for their different goals and knowledge, to model theirs. Again, the model involves a range of hypotheses about things they could do but eventually choose not to. Our model of the world includes 'free will'.
The process of taking a model of the world, exploring all the possible alternative inputs (things we don't know, future actions), and picking the best of the outcomes can be done as a deterministic algorithm. We loop through the variables, evaluate each against our desires and goals, and deterministically select the one that works best. But this deterministic algorithm contains representations of counterfactual alternatives and agents with free will, which (depending on how you define them) in a deterministic universe are impossible.
So in this sense, in a deterministic universe there are certain features of our reason and logic that are illusory. But the reason and logic themselves are not. They still exist, and function. They still work very successfully, in the evolutionary sense of giving answers accurate and reliable enough for survival. But they are only a crude approximation to reality, ones that only work most of the time, but can give false answers if pushed beyond their domain of validity.
The debate goes far beyond that simple picture. There are alternative ways to define terms to make determinism compatible with free will. You can find further information on the debate on Compatibilism starting here.