[Fire HD 8 and HD 10 General] GoodSoul:
Here is what I did, and it worked perfectly:

adb push com.amazon.unifiedsharefacebook.apk /storage/sdcard0/
adb shell
pm install -r -d /storage/sdcard0/com.amazon.unifiedsharefacebook.apk
run-as com.amazon.unifiedsharefacebook

pm hide com.amazon.accessorynotifier
pm hide amazon.jackson19
pm hide com.amazon.mw.sdk
pm hide com.amazon.mw
pm hide com.android.email
pm hide org.mopria.printplugin

pm hide com.amazon.kindle.unifiedSearch
pm hide com.amazon.ods.kindleconnect
pm hide com.amazon.photos
pm hide com.amazon.unifiedsharegoodreads
pm hide com.amazon.unifiedsharesinaweibo
pm hide com.amazon.unifiedsharetwitter
pm hide com.amazon.venezia
pm hide com.amazon.webview.metrics.service
pm hide jp.co.omronsoft.iwnnime.languagepack.zhcn_az
pm hide jp.co.omronsoft.iwnnime.mlaz

pm hide org.mopria.printplugin
pm hide amazon.jackson19
pm hide com.amazon.accessorynotifier
pm hide com.android.email

pm hide com.amazon.advertisingidsettings
pm hide com.amazon.ags.app
pm hide com.amazon.calculator
pm hide com.amazon.client.metrics
pm hide com.amazon.cloud9
pm hide com.amazon.cloud9.kids
pm hide com.amazon.dp.contacts
pm hide com.amazon.dp.fbcontacts
pm hide com.amazon.geo.client.maps
pm hide com.amazon.kcp.tutorial
pm hide com.amazon.kindle.kso
pm hide com.amazon.kindle.otter.oobe
pm hide com.amazon.kindle.otter.oobe.forced.ota
pm hide com.amazon.kindle.personal_video
pm hide com.amazon.kor.demo
pm hide com.amazon.mp3
pm hide com.amazon.parentalcontrols
pm hide com.amazon.photos.importer
pm hide com.amazon.pm
pm hide com.amazon.precog
pm hide com.amazon.socialplatform
pm hide com.amazon.tahoe
pm hide com.amazon.weather
pm hide com.amazon.webview
pm hide com.android.calendar
pm hide com.android.contacts
pm hide com.android.deskclock
pm hide com.android.music
pm hide com.android.onetimeinitializer
pm hide com.android.protips
pm hide com.kingsoft.office.amz

exit

reboot

Fire OS: 5.6.2.0

Device: Fire HD 8 (7. Generation)

Now I am ad free. :D

After executing above statements you will be unable to enter the first three options under "personal" in the settings. But that's just amazon crap.

I also replaced my launcher with Nova Launcher (using Launcher HiJack) before entering above commands. That might be important!

Thanks to [MENTION=7935372]Sindee75[/MENTION] - Please note, if you are using the amazon appstore/store then do not hide com.amazon.venezia, else you might run into problems.

Update:

In case that you do not want to be auto-updated you can also run the following commands before the exit from the script above:

pm hide com.amazon.settings.systemupdates
pm hide com.amazon.device.software.ota

Booting is now a little bit faster (bcs. no so much crap is loaded anymore). Overall the tablet feels a lot smoother and more flexible. The ads on the lookscreen are gone - even after a reboot (that was an issue before). Due to the reason that a lot less apps start on boot time the sdcard gets available a lot faster (at least on my system). Overall, after opening the first apps after booting, if feels faster and faster the longer you use it. But this is only my experience.

TWIMC: How to install Google Play / Nova Launcher

If you do not have any "Kindle eBooks" from Amazon you can hide Kindle as well:

com.amazon.firelauncher