Some CSS fixes to make SharePoint 2010 Publishing sites get along better with Bootstrap 3. For the most part, it sets SP2010's own UI elements to use `box-sizing: content-box;` instead of `box-sizing: border-box;` as used by BS3. Either import this at the end of bootstrap.less, or after importing bootstrap.less into your own .less files.

SP2010-Bootstrap3-Fixes.less

/****************************************************************************
 *
 * This LESS file contains fixes to make Bootstrap 3 play nicely with
 * SharePoint 2010 Publishing pages. It mainly switches SP2010's UI 
 * elements back to the "content-box" box-model. 
 *
 * It also incorporates the CSS portions of Kyle Schaeffer's fixes for
 * scrolling on SP2010. The JS part is called from the master page.
 *
 ****************************************************************************/

/****************************************************************************
 *
 * BEGIN Fixes for scrolling on SP2010
 * See: http://kyleschaeffer.com/sharepoint/sharepoint-2010-scrolling/
 * Note: The JS patch is called from the master page.
 *
 ****************************************************************************/

html {
    min-height: 0;
}

body {
    #s4-ribbonrow {
        position: fixed;
        z-index: 1000; /* Override Bootstrap */
        .ms-MenuUIPopupBody,
        .ms-popoutMenu,
        .ms-cui-menu[id ^= "Ribbon."] {
            position: fixed !important;
        }
    }
    #s4-workspace {
        overflow: visible !important;
    }
    form#aspnetForm {
        height: auto;
    }
}
.navbar-default {
    border-width: 0;
}
.ms-dlgOverlay {
    width: 100% !important;
}

/****************************************************************************
 *
 * END   Fixes for scrolling on SP2010
 *
 ****************************************************************************/

/****************************************************************************
 *
 * BEGIN Sharepoint 2010 Fixes for Bootstrap 3
 *
 ****************************************************************************/

.sbg-contentboxmodel {
    -webkit-box-sizing: content-box;
    -moz-box-sizing: content-box;
    box-sizing: content-box;
}

/* Allows the site to be responsive */
body #s4-bodyContainer {
    min-width: 0;
}

/* Fixes a bit of ribbon breakage */
body #s4-ribbonrow {
    line-height: 1.2;
    .s4-breadcrumb-menu {
        background-color: #FFFFFF;
    }
    a:hover,
    a:focus,
    a:active {
        text-decoration: none;
    }
}

/* Fix some text sizing issues with the .ms-dlg modal popup, tooltips and the page status bar */
body {
    #pageStatusBar,
    .ms-cui-tooltip {
        line-height: 1.1;
        font-size: 8pt;
    }
    .ms-dlgContent {
        line-height: 1.1;
    }

    /* There are a lot of things in the SP2010 UI that don't like box-sizing: border-box */
    #s4-ribbonrow *,
    #s4-ribbonrow *:before,
    #s4-ribbonrow *:after,
    *[class*='ms-dlg'],
    *[class*='ms-dlg']:before,
    *[class*='ms-dlg']:after,
    .ms-dlgFrameContainer > div,
    .ms-dlgFrameContainer > div:before,
    .ms-dlgFrameContainer > div:after,
    .ms-dlgFrameContainer > div > div,
    .ms-dlgFrameContainer > div > div:before,
    .ms-dlgFrameContainer > div > div:after,
    .ms-MenuUIPopupBody,
    .ms-MenuUIPopupBody:before,
    .ms-MenuUIPopupBody:after,
    .ms-MenuUIPopupBody *,
    .ms-MenuUIPopupBody *:before,
    .ms-MenuUIPopupBody *:after,
    .ms-ToolPaneOuter,
    .ms-ToolPaneOuter:before,
    .ms-ToolPaneOuter:after,
    .ms-ToolPaneOuter *,
    .ms-ToolPaneOuter *:before,
    .ms-ToolPaneOuter *:after,
    *[class*='ms-cui'],
    *[class*='ms-cui']:before,
    *[class*='ms-cui']:after,
    *[class*='ms-cui'] *,
    *[class*='ms-cui'] *:before,
    *[class*='ms-cui'] *:after {
        .sbg-contentboxmodel;
    }
    /* Work around an issue in IE10, where drop-down menus blank the screen 
       See http://davemcmahon81.wordpress.com/2012/12/11/sharepoint-2010-ribbon-and-ie10-screen-goes-white/
    */
    .ms-cui-modalDiv-ie,
    .ms-cui-glass-ie {
        background-color: transparent;
    }
}
/* Note I added class="outer" to <div id="MSO_ContentDiv"> in the Master Page */
.outer {
    /* Sort out some font size and face issues (overrides corev4.css:1072) */
    .ms-WPBody {
        font-family: inherit;
        font-size: inherit;
    }
    /* Responsive images break the webpart toolpane */
    .ms-ToolPaneOuter img {
        display: inline-block;
        max-width: none;
    }
}

/****************************************************************************
 *
 * END   Sharepoint 2010 Fixes for Bootstrap 3
 *
 ****************************************************************************/