@TypeIA's answer is correct, but I thought it might be worth clarifying.
Both 'Mayor' and 'Wildlife Liaison Officer' are part of the person's title in the context used. They therefore should not be separated by commas. They are, in effect, part of the person's name.
Mayor John Smith
Wildlife Liaison Officer Jane Brown
Or, to use an example from Catch 22:
Major Major Major Major.
If an article were to be used in any of those sentences, then a comma becomes required, as the person's name then acts to qualify the noun, and is not a required part of the sentence.
The Wildlife Liaison Officer, Jane Brown, said that ...
Bonus off-topic mumbling:
In Canada it is common in modern practice to refer to MPs and provincial MPPs as if MP/MPP were a title conveying honour and not simply a post-nominal description of someone's job. "MP Mary Jones", for example.
This practice annoys me greatly. In the UK the usage certainly used to be to say, for example, "Mr John Smith, MP", and even "the Prime Minister, Mrs Margaret Thatcher". This distinction, and the use of a comma, emphasized that politicians were still entirely ordinary people and did not gain a special titles by election. It sounded jarring when Americans would say things like "Mr Prime Minister" or "Prime Minister Thatcher". Worse still was any suggestion of continuing to call someone "Prime Minister X" after they had left office.
It's a subtle point, but the use of commas and articles in these cases makes an important egalitarian distinction that I for one would prefer to see continue.