Running Drupal 6 with PHP 5.3 and Maintaining Security
Here are the patches I'm using for D6. Some of these issues will have been committed, thus they don't need a patch anymore.

Views
http://drupal.org/node/853864#comment-6465560 - views_get_default_view() - race conditions and memory usage
http://drupal.org/node/402944#comment-4650020 - Persistent caching for unpack_options() calls from building displays
http://drupal.org/node/1372388#comment-5370432 - Inline/Remove recursion in views_object->unpack_options to improve performance
http://drupal.org/node/465332#comment-4756670 - Strict warning: non-static method should not be called statically
http://drupal.org/node/893128#comment-4894754 - Strict warning: Declaration of * should be compatible with that of views_handler::*
http://drupal.org/node/1265138#comment-4928988 - Notice: Undefined variable: teaser in views_plugin_row_node_rss->render()
http://drupal.org/node/1432242#comment-5807802 - Notice: Undefined index: size in views_db_object->export_row()  

Views Bonus Pack
http://drupal.org/node/1223910#comment-4756506 - Strict warning: is_a(): Deprecated

Views Bulk Operations
http://drupal.org/node/1782662#comment-6469236 - Notice: Undefined index: HTTP_USER_AGENT in views_bulk_operations_form()

Views Datasource
http://drupal.org/node/1265190#comment-4929150 - Notice: Undefined property: stdClass::$name in views_plugin_style_xml->option_definition()

Views Slideshow JCarousel
http://drupal.org/node/1885586#comment-6926004 - php notices



Here are some of the more popular modules as well

Calendar
http://drupal.org/node/999514#comment-4761150 - Strict warning errors for views plugins

CCK
http://drupal.org/node/1118792#comment-4358322 - array_keys() expects parameter 1 to be array, null given in content_multigroup.node_form.inc
http://drupal.org/node/1244534#comment-4845816 - Notice: Undefined index
http://drupal.org/node/1247732#comment-4857954 - Undefined index: _remove
http://drupal.org/node/1156096#comment-4868230 - Notice: Undefined index: title/description in content_storage()
http://drupal.org/node/1251146#comment-4871780 - Notice: Undefined index: #children in template_preprocess_content_field()
http://drupal.org/node/1251160#comment-4871816 - Notice: Undefined index: #field_info in fieldgroup_form_alter()
http://drupal.org/node/1265080#comment-4928778 - Notice: Undefined property: stdClass::$broken in content_handler_field->render()
http://drupal.org/node/1316510 - fix undefined index '#id' notice in form_alter hooks
http://drupal.org/node/533610#comment-5157454 - Notice: Undefined index: safe in theme_text_formatter_default()
http://drupal.org/node/1762506#comment-6411880 - Notice: Undefined property: stdClass::$field_* in content_rules_field_has_value()  

CTools
http://drupal.org/node/1226098#comment-4765262 - Strict warning: Only variables should be passed by reference
http://drupal.org/node/1334894#comment-5215962 - Warning: Invalid argument supplied for foreach() in views_content_views_content_type_render()
http://drupal.org/node/1510118#comment-5807950 - Strict warning: Creating default object from empty value in views_content_views_content_type_admin_info()  

Context
http://drupal.org/node/866208#comment-4845268 - Notice: Undefined property: stdClass::$taxonomy in context_node_condition() on line 352  

Date
http://drupal.org/node/802722 - Extra comma in output when using "display from and to dates" on Display fields configuration
http://drupal.org/node/1223956#comment-4756746 - Strict warning: Creating default object from empty value in date_handler_fields
http://drupal.org/node/1948238#comment-7200236 - Strict warning: Declaration of date_handler_field_multiple::pre_render()   

Features
http://drupal.org/node/1516900#comment-5833834 - Notice: Undefined index: group in views_handler_dependencies()
http://drupal.org/node/1515878#comment-5833794 - Strict warning: Only variables should be passed by reference in content_features_rebuild()   

FileField
http://drupal.org/node/1180508#comment-4872304 - Warning: array_merge() Argument #2 is not an array in filefield_widget()
http://drupal.org/node/1164892#comment-4500360 - Warning: Trying to get property of non-object in filefield_field_sanitize()
http://drupal.org/node/1317424#comment-5146998 - Notice: Undefined index: fid in filefield_widget_process
http://drupal.org/node/1415840#comment-5514392 - Fatal error: Cannot unset string offsets
http://drupal.org/node/1677208#comment-6213652 - Notice: Undefined index: data in theme_filefield_widget_preview
http://drupal.org/node/1775708#comment-6440138 - Notice: unserialize(): Error at offset 0 of 1 bytes in filefield_get_node_files()  

ImageAPI
http://drupal.org/node/1189884#comment-4608576 - Allow for custom code to be inserted into the convert command before it is ran
http://drupal.org/node/1201914#comment-4663120 - Add in the ability to "nice" the convert process
http://drupal.org/node/1250348#comment-4868656 - Fix divide by zero errors.
http://drupal.org/node/1766064#comment-6434498 - User error: ImageMagick reported error code 1. Message: convert: no encode delegate for this image format   

Imagecache
http://drupal.org/node/1243258#comment-4841244 - use lock.inc instead of a file lock
http://drupal.org/node/1761108#comment-6407780 - Notice: Undefined index: filepath in theme_imagecache_formatter_default()  

jQuery Update
http://drupal.org/node/1067290#comment-6516884 - Fix jQuery 1.7 for Drupal 6  

Rules
http://drupal.org/node/1250384#comment-4868884 - Strict warning: Only variables should be passed by reference in rules_core_action_execute()   



Core patches
http://drupal.org/node/496184#comment-5787602 - db_set_active errors out if db_connect fails. Add in the ability to handle this gracefully (don't always call _db_error_page)
http://drupal.org/node/764726#comment-5787642 - (Shared taxonomy patch) hook_taxonomy_term_presave() is missing
http://drupal.org/node/905156#comment-5791724 - PHP notice in form_builder function
http://drupal.org/node/1006938#comment-4661790 - Remove PHP warnings from legacy PHP4 XML parser for new update status release history XML
http://drupal.org/node/352180#comment-2562140 - Better, multi-site friendly "www." addition/removal in .htaccess
http://drupal.org/node/939810#comment-5791876 - Notice: Undefined index: key in format_xml_elements()
http://drupal.org/node/985814#comment-5791916 - Undefined index: configurable in actions_synchronize()
http://drupal.org/node/557542#comment-4989990 - Cache module_implements()
http://drupal.org/node/1317236#comment-5146174 - Race condition in file_delete: Warning: unlink() No such file or directory in file_delete()
http://drupal.org/node/1004820#comment-5171678 - Notice: Undefined index: type in dblog_build_filter_query()
http://drupal.org/node/1345204#comment-5259212 - Optimize element_children()
http://drupal.org/node/753064#comment-5274348 - _menu_link_translate() might avoid calling _menu_load_objects()
http://drupal.org/node/261148#comment-5798704 - menu_masks variable is empty (race condition)
http://drupal.org/node/1688282#comment-6240944 - Unknown error: Function split() is deprecated in _filter_autop()
http://drupal.org/node/521838#comment-6375266 - Clean up drupal_get_schema_versions()
http://drupal.org/node/1710656#comment-6474030 - If item is hidden in _menu_tree_check_access() skip it right away.
http://drupal.org/node/1882556#comment-6913038 - Notice: Undefined index: REMOTE_ADDR in includes/bootstrap.inc on line 2835
https://www.drupal.org/node/2292125#comment-8912499 - PHP 5.5 - preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in decode_entities() (line 349 of includes/unicode.inc)  



Most of the issues I'm following that have the word notice in it
https://www.drupal.org/project/issues/search?text=notice&amp;projects=&amp;project_issue_followers=mikeytown2&amp;status[0]=Open&amp;categories[0]=1&amp;issue_tags_op=%3D&amp;assigned=&amp;submitted=&amp;issue_tags=&amp;&amp;order=field_issue_version&amp;sort=asc

And the ones that contain the word warning
https://www.drupal.org/project/issues/search?text=warning&amp;projects=&amp;project_issue_followers=mikeytown2&amp;status[0]=Open&amp;categories[0]=1&amp;issue_tags_op=%3D&amp;assigned=&amp;submitted=&amp;issue_tags=&amp;&amp;order=field_issue_version&amp;sort=asc



Query used to find the worst offenders for php issues: http://groups.drupal.org/node/170339