Should I remove them. I've got a blog with evergreen content.
Dates within the content are really not used. Exceptions in the past has been when a pages creation and modification dates cannot be determined through ordinary means. Google, for example, gets it's creation date from the date the page was discovered. If the page changes and the changes are not superficial, then Google will note that as a modification date.
From a user standpoint, it is best to have the date an article was written and another date for when the article was last updated. These are good things. I look for them especially for topics that are dated such as SEO. I cannot tell you how many SEO articles are out there that are just plain wrong because the world of search changes so rapidly. It helps the user to determine if the article still applies. Keep in mind that sometimes people want older articles. I know I do. It helps with research. As well, sometimes it is the earlier written articles that are the most accurate- especially for certain topics. Earlier articles often have better details. For example, in SEO, many article were written by "me too" bloggers that do not offer anything new while the older original articles tend to explain how search works in simpler terms that largely have not changed. The "me too" articles tend to be shallow and misleading without historical context. This is especially true for SEO.
The decision to have dates is purely up to you. If it is not too much of a pain, I would continue to use them to help your audience. You can also link to other newer article and vice versa.
Adding to what closetnoc already said, we can take 2 examples directly from Google: https://developers.google.com/structured-data/?rd=1
Look at the bottom. The structured data
itemprop="datePublished" wraps the Last Updated Date, not the date of when it was first published (it's not even there).
Examples of when freshness matters and when not
If I search for [olympics], I probably want information about next summer’s upcoming Olympics, not the 1900 Summer Olympics (the only time my favorite sport, cricket, was played). Google Search uses a freshness algorithm, designed to give you the most up-to-date results, so even when I just type [olympics] without specifying 2012, I still find what I’m looking for.
There are plenty of cases where results that are a few years old might still be useful for you. [fast tomato sauce recipe] certainly saved me after a call from my wife reminded me I had volunteered to make dinner! On the other hand, when I search for the [49ers score], a result that is a week old might be too old.
In this blog post, you can see that instead the structured data wraps the original date, and last update is just written without special markup.
I won't go in depth there's people that can explain that to you better than me.
That said, I would not remove Blog Post Date. In case it's irrelevant for Google, maybe (probably) it could be for the user, and you should think about them.
Here's a similar topic on stackexchange How does Google recognize publish date of a post
You should keep the dates in some way or another, even for "evergreen" content. I never figured out how (its not header, schema, nor meta), but certain SaaS are able to push a modified date to Google SERPS resulting in a way fresh date indicator. Maybe its RSS somehow, but I dont think...it must be similar structure to so many sites. Anyways, even if the content never changes, this SERP indicator hints to users that the content maintained recently.
At the least, even without date SERP widget, and not displaying a visual date to users, keeping the content groomed and fresh would most def trigger the same result. Care about your content like you would groom an orchard, prune out the dead stuff on those trees, see where the dead fruit is :)
From an opinionated angle, I prefer to always see dates on most content from which I learn anything. This is especially true in arenas where concepts, tools, recommendations, etc can become obsolete. Call it nerdy, but having the original post date + the modified on date tactfully put somewhere in the page is an asset not a downfall.
In the end, my answer is leave it in.