Because climate change is not natural, must it therefore be bad?

This question is so full of fallacies it borders on fraud. Let's start with the most obvious, misstating the question. The problem in the worldwide debate is anthropogenic global warming, or AGW. During the Obama years, after a couple of years where climate warming leveled off, many people changed that to "climate change" so as to be less scary, which was a mistake. I believe this falls under the fallacy called the "loaded question".

Then there is the "appeal to nature" fallacy, in which saying that because something is natural means that is is therefore valid, justified, inevitable, good, or ideal. Assuming you believe it is natural.

Next, saying climate change is natural because it has happened before is cherry picking or the "Texas Sharpshooter". I'm not sure what fallacy would describe completely ignoring all evidence to the contrary of your opinion. But we know from scientific evidence that warming periods in the past occurred over hundreds or thousands of years, but this change is happening over about a hundred, and is accelerating. And we know that increases in CO2 and methane cause global warming, and we know that we have overloaded the atmosphere with massive increases in these gases over the period in which the average global temperature has been rising. Putting it all together, through scientific modeling, we know from evidence that to say AGW is "natural" is not just false, but dishonest.

Parallel to this last fallacy is the "Gambler's Fallacy". This is where some point to brief periods of variation in which the temperature does not rise, or even falls, as evidence that rising trends are false.

Then the question is, if it is unnatural, is it "bad"? That is a "Strawman" if I ever saw one, along with "Moving the Goalposts". The science predicts that AGW will result in unbearable temperatures, rising sea levels and coastal flooding, increases in catastrophic losses from floods, hurricanes, fires, and drought, shortages of food, increases in diseases, a massive die-off and extinction of many species of plants and animals, and many other disastrous consequences. All of which is now being verified by our common experience worldwide, increasingly every year. To ask us to knock down the claim that it is "bad" is fallacious itself. If you can't conclude that these changes are going to be "bad", you should stay out of the discussion.

There is so much wrong with this "asked and answered" question I almost declined to comment. But these are the classic talking points among AGW deniers, so they must be debunked aggressively. Others of the most typical logical fallacies applied to AGW are the "Anecdotal" fallacy ("it's snowing outside, what global warming?), the Bandwagon fallacy ("millions of people are skeptical about AGW"), and the Slippery Slope ("you won't be able to fly, or eat meat"). I went through the list of fallacies and can think of example of almost every one used to deny global warming. This question should go into the Logical Fallacy Hall of Fame.

Climate change refers to changes which are directly caused by people. Yes, there have been various climates in the past, but that says nothing about the fact of climate change.

Diamonds form in nature, does that mean that synthetic diamonds aren't real?

One perspective of climate change being bad is that large populated areas will become submerged. Places like Bangladesh will flood, there will be migrations of refugees trying to find places to go while resources are becoming scarcer, and this will likely lead to armed conflicts.

You might say that this is from the perspective of humans, but all such discussions are front that perspective. Also all ecosystems would change, maybe with the except of the poles, and all species would be challenged to survive. Where's the good perspective? If you think humanity needs wiped out then perhaps, but that's a fringe view and not part of policy making.

Edit: actually I didn't address your opening question of because climate change is unnatural it must be bad. Sure, that would be a fallacy, there are lots of things which aren't natural which are considered good. I would, however, question whether that's a real argument. Most people say that it's bad because of all the extreme weather, sea level rise, viability of growing  the same crops, extinction, etc.

The Appeal to Nature is common because there are many who claim that nature is good, and not nature is bad. To claim something is bad because it is natural is uncommon, and would simply fall under the generic Non Sequitur .

To argue that climate change is unnatural is a fallacy.

I don't see how. That would be a truth claim. Of course, we need to know what is meant by "climate change" as well as "unnatural."

To argue that it is bad is another question: it would depend on one's perspective. 

Sure. But some perspectives are easier to argue than others.