These are the most common things I see in published visualizations, all of which I've done at some point. While there's nothing inherently wrong with any of these things, sometimes they detract from the story you're trying to tell. Take the time to polish your viz because putting on those finishing touches might get you that job one day.

Most of the examples I'm showing you are from a viz of my own that I broke. Examples that came from elsewhere are given attribution in the caption of the image. I've also included a few links in each section that might give you the how-to that I don't detail here.

Publishing selected (unintentional) filters and marks

We've all done it - accidentally left a mark or a filter selected when we published the viz - only to realize it after you see it on Tableau Public. Leaving it selected (turned on) could suggest to your reader that you want her to focus on that mark or filter. To fix this, go back into Tableau, de-select and re-publish. Tableau Public won't 'reset' your viz for you - it assumes you want to focus on that item.

Starting on the wrong page

You've put in all this work, and you just want to publish your viz and get dinner. Don't forget: your viz will publish on the page (red box) you were looking at when you published, even if you took the time to create an Intro page (green box). Make sure you publish the correct 'landing' page.

Displaying sheets you don't intend others to see

If you want your readers to interact with a dashboard or story (and not the individual sheets), then you have two choices:

Hide all of the sheets before publishing and make sure you're on the story or dashboard page upon publishing. You can even hide unused sheets on a placeholder dashboard, if that helps.
Don't hide the sheets, make sure you're on the story or dashboard page you want them to see and don't select 'Show workbook sheets as tabs' in the Edit Details button on Tableau Public.

Forgetting to title your work, give context, or provide attribution for your data sources

Ever see a viz and think to yourself, 'what am I supposed to be looking at here?' or 'what is this data?' Don't let this happen to you.

Title your visualizations, even if your title is a simple question like 'How many 5-star ratings did the top 25 books get?'

If the title isn't clear enough, provide context with a couple of sentences at the top of the chart. Or include an Info button that displays additional detail via tool tip.

And finally, give credit to the sources of your data. Always. If the data is private to a company (but they provided it to you with permission), say so - 'Used with permission from X company.'

Using dual axis charts questionably

Image comes from Mother Jones: http://www.motherjones.com/kevin-drum/2015/09/lying-charts-anti-abortion-edition/

If your data requires you use a dual axis chart, then use it correctly. If you aren't going to synchronize your axes, then make sure your readers understand they might be looking at a difference in the measures. One suggestion if you're not synchronizing axes: color the axis to match its corresponding mark - that way there's less debate about which axis applies to which mark in the chart.

Here are some good links about dual axis charts:

Thinking variety is what your viz needs

They say less is more for a reason. Too much of anything isn't variety - it's confusing, and can blind your reader to the story you're trying to tell. Exercise restraint.

Colors
A chart (or dashboard) with too many colors makes it difficult for the reader to recall the subtle value differences between the colors, especially if the same colors are used differently on different charts. Try picking a color palette for your topic, or design one with a few complementary colors in a tool like Paletton. Here are some useful links about colors:
Custom shapes
Folks like to use brand logos as custom shapes, and I cheer that until it gets hard to read (or the reader isn't familiar with the logos). Custom shapes - even the simple ones that come with Tableau (circles, diamonds, triangles, etc.) - become another layer of data the reader needs to 'translate' before understanding. Combine that with too many colors and you make things even more difficult to read. Some links to read more about custom shapes are below.
Fonts
I'm all for a fun font in the title of the viz, but that's the only place where I want to see it. Can you imagine reading your Kindle with all-script fonts? I don't think so.

Thinking auto-play is clever

I'm not going to include an example of this because, well - every time I open this post, it would start to play music.

Auto-playing music and video are fine if you start on mute and allow your reader to turn up the volume. But auto-playing music or video, especially when there's nothing to prepare your reader for the volume that might come out of their computers (at work), can be jarring. When I accidentally open one of these, I scramble desperately for the mute button on the viz (it's never obvious) or start jamming my volume button on my computer.

If you want to introduce music, follow Skyler Johnson's (here and here) and Adam McCann's (here) expert examples and allow the user to press play on her own. Don't let auto-play become the scrolling marquee of our future.

More tips:

Forgetting to adjust the size of your viz to eliminate scroll bars

In the image above, there are two things happening.

The chart with the pie charts doesn't fit in the container I've put it in, so it has scrollbars. Fix this before publishing by clicking the carat in the upper right-corner of the chart and choosing Fit, Entire View (or Fit Width or Fit Height - you make the right selection).
The dashboard doesn't fit the story layout. By default, the storypoints canvas is 1016x964. A dashboard defaults to 1000x800. If you adjust nothing, the dashboard will fit onto the story. But if you make the dashboard really tall, then it will display scroll bars on the story.

Don't forget to check for scrollbars throughout your finished work.

More sizing tips:

Forgetting to format your tool tips

Take the time to format your tool tips, or if they're not needed turn them off or disable the command buttons so the little toolbar above the tool tip doesn't appear. This seems like such a little thing, but I guarantee it's one of the most noticeable.

More tool tip tips:

Forgetting to polish your map

In addition to charts needing polish, maps have very particular functionality that only gets better with a little thought. When working with maps, don't forget to think about:

Changing the map layers so the correct level of detail shows up for the zoom level of the map. There's no reason to show county or state names if you're looking at city level data. If you're looking at city level data, consider showing street names.
Changing the map background. There are 2 other options (dark, normal) in the map layers menu, and a host of free backgrounds using a Mapbox token (built in to Tableau).
Fixing the map focus and removing the map toolbar. This can prevent inadvertent shifting of the map.
Resolving the NULL locations ore removing the indicator altogether.
Remembering the colors, shapes and tool tips suggestions in general. These all apply to maps.
Using maps intentionally. Don't just use them because you have geographic data. Use them because it's important to the story you're telling.

More maps tips:

All of the items here are suggestions, not rules. There will be plenty of occasions when you intentionally choose a path that's contrary to what I've suggested - and that's the point - you're being intentional about it. Refer to this list until you've developed the muscle to do these things automatically - and your portfolio will look better for it.

Viz Polishing Checklist - Helpful Tips