Is the effectiveness of chemical catalysts affected by temperature the same way enzymes are?
This is a somewhat complex question to answer because even your ‘simple’ enzyme example isn’t as simple as it looks. Let me discuss enzyme optimum temperatures first.

For any reaction that is exergonic — and that explicitly includes enzyme catalysis — an increase in temperature raises the reaction rate. This is due to at least two microscopic features:


higher temperature means more and faster movement thus a higher probability of collisions
each collision has a greater chance of being energetic enough to supply the required activation energy


This is condensed in van ’t Hoff’s postulate that an increase of the reaction temperature by $10~\mathrm{^\circ C}$ leads to an increase of the reaction rate by a factor of $2$ to $4$. Therefore, we would initially expect all enzymatic processes to be faster with increased temperatures.

However, the enzyme itself is a highly fragile system that is held together a good deal by noncovalent interactions. These can break apart under elevated temperatures causing the well-balanced enzymatic structure to break down (‘denature’)  into a seemingly random protein chain. This also breaks the catalytic centre meaning that the enzyme can no longer catalyse efficiently. Taking these two effects together, we arrive at the optimum curve of enzyme catalysis.



Chemical catalysts come in a number of shapes and varieties. Consider for example:


the catalyst (mainly platinum as I heard at school) in your car’s exhaust
the complex catalytic oxide system in the Haber-Bosch process
transition metal catalysts such as $\ce{[Pd(PPh3)4]}$
acid catalysis, i.e. dissolved $\ce{H+}$ is the catalyst
organic catalysis
…


Most of these will have some equivalent of a denaturation temperature; e.g. once platinum metal (or palladium, if on charcoal for typical organic hydrogenations) melts, it can no longer act as a catalyst for the reaction in the way it could in solid state. Thus, there should be a ‘denaturation equivalent temperature’ for most chemical catalyst systems at some point. Some compounds — the Haber-Bosch catalysts jump to mind — will, however, probably melt at such high temperatures that you would need to live on the sun’s surface for that to happen. Therefore, we could say that no limit exists for all intents and purposes.

Catalysts such as tetrakis(triphenylphosphane)palladium(0) need to dissociate at least one ligand to be catalytically active. An increase in temperature should enhance that dissociation. Therefore, many homogenously palladium-catalysed reactions are indeed sped up by a moderate increase in temperature. However, side reactions also speed up.

Organocatalysts and simple proton catalysis should be among the most temperature-insensitive — the former because all bonds should be covalent, the latter because it is just a reactive, activating species that cannot be poisoned. However, these two require a medium in which they act; the boiling point of the solvent is the highest possible temperature for their reactions.



Thus basically: yes, all chemical catalysts have some maximum useable temperature somewhere, but for most it does not matter for practical intents and purposes. Also, there are no simple models to explain this like in the enzyme case, hence it isn’t typically taught.