Idea from T34603: search around MediaWiki for "remove after 1.XX release" code and similar and kill that code (or file bugs to remember to remove the code later).
Make sure you use a commit summary that explains what you're doing/removing (reusing this task title isn't a good one!), and also add an entry to RELEASE-NOTES-1.29 similar to those that exist already https://github.com/wikimedia/mediawiki/blob/2c74b04/RELEASE-NOTES-1.29#L54-L68
Example: https://www.mediawiki.org/wiki/Special:Code/MediaWiki/104318 added "Do not release 1.19 with this." in a comment. Search for phrases like that.
List of things that could be removed (thanks to @Matthewrbowker):
**Remove after...**
<None found>
**Deprecated, use ... instead**
* class ResetUserTokens
* Skin::doEditSectionLink()
* User::getImplicitGroups()
**@deprecated (Soft deprecated)**
* class ProfileSection
* function wfProfileIn() (ProfilerFunctions.php)
* function wfProfileOut() (ProfilerFunctions.php)
* class SpecialResetTokens
* EDIT_TOKEN_SUFFIX (in User.php)
* User::randomPassword()
* User::setPassword()
* User::setInternalPassword()
* User::setNewpassword()
* User::getTokenFromOption()
* User::setCookie()
* User::clearCookie()
* User::setExtendedLoginCookie()
* User::checkPassword()
* User::checkTemporaryPassword()
* User::getEditTokenTimestamp()
* User::addNewUserLogEntry()
* User::addNewUserLogEntryAutoCreate()
* BufferingStatsdDataFactory::getBuffer()
* User::getPasswordFactory()
* User::passwordChangeInputAttribs()
* Language::initEncoding()
* Language::recodeForEdit()
* Language::recodeInput()
* Interwiki::isValidInterwiki()
* Interwiki::fetch()
* Inetwiki::getAllPrefixes()
* class AuthManagerAuthPlugin
* Class AuthManagerAuthPluginUser
* Status::getErrorsArray()
* Status::getWarningsArray()
* $mUniqPrefix (in class Parser)
* Parser::getRandomString()
* Parser::uniqPrefix()
* Parser::disableCache()
* SkinFactory::getDefaultInstance()
* ApiQueryInfo::getTokenFunctions()
* ApiQueryInfo::resetTokenCache()
* ApiQueryInfo::getEditToken()
* ApiQueryInfo::getDeleteToken()
* ApiQueryInfo::getProtectToken()
* ApiQueryInfo::getMoveToken()
* ApiQueryInfo::getBlockToken()
* ApiQueryInfo::getUnblockToken()
* ApiQueryInfo::getEmailToken()
* ApiQueryInfo::getImportToken()
* ApiQueryInfo::getWatchToken()
* ApiQueryInfo::getOptionsToken()
* LoadBalancer::getLaggedSlaveMode()
* LoadBalancer::laggedSlaveUsed()
* MimeMagic::singleton()
* class ApiTokens
**wfDeprecated (Hard deprecated)**
* Parser::getRandomString()
* Parser::uniqPrefix()
* $uniq_prefix in Parser::extractTagsAndParams()
* ApiQueryLogEvents::addLogParams()
* In DifferenceEngine::textDiff() there is a notation deprecating a variable
* $wgSessionsInObjectCache in class Setup
* SearchEngine::getOpenSearchTemplate()
* WikiImproter::__construct() without a Config instance
* LoginSignupSpecialPage::getFakeTemplate() has hook references depricated
* ChangeTags::listExtensionActivatedTags()
* ChangeTags::listExtensionDefinedTags()
* VFormHTMLForm::isVForm()
* User::setCookie()
* User::clearCookie()
* User::setExtendedLoginCookie()
* ApiPageSet::getInvalidTitles()
* User::createNew()
* User::getEditTokenTimestamp()
* User::getPasswordFactory()
* Revision::getRawUser()
* Revision::getRawUserText()
* Revision::getRawComment()
* MWLBFactory::getLBFactoryClass() (inside the class)
* Class LoginForm
* Job::batchInsert()
* UploadBase::stashFileGetKey()
* UploadBase::stashSession()
* MWExceptionHandler::getLogId()
* ApiResult::__construct - handing instance of ApiMain
* ApiBase::getModuleProfileName()
* ApiBase::safeProfileOut()
* ApiBase::getProfileTime()
* ApiBase::profileDBIn()
* ApiBase::profileDBOut()
* ApiBase::getProfileDBTime()
* ResourceLoader::readStyleFiles() without a ResourceLoader context.
* SpecialPageFactory::getList()
* SpecialPageFactory::getPage() - Array syntax
* StripState::__construct called with any arguements
* GlobalFunctions.php - wfErrorLog()
* GlobalFuctions.php - wfFixSessionID()
* GlobalFunctions.php - wfResetSessionID()
* GlobalFunctions.php - wfSetupSession()
* Linker::getLinkColour()
* Linker::link() with $query as string
* SkinTemplate::outputPage() (Not sure on this one, it's hidden in an if())
* Linker::formatTemplates()
* MailAddress::__construct() called with a user object
* Block::__construct with multiple arguements
* OutputPage::__construct with no context
* OutputPage::addExtensionStyle()
* OutputPage::getExtStyle()
* HTMLForm::isVForm()