You can use the method CookieManager#setCookie(String url, String value). As stated in the docs:
Sets a cookie for the given URL. Any existing cookie with the same host, path and name will be replaced with the new cookie.
The "clearest" way is to set all cookies created by twitter to expired (a time in the past). The code from this answer is almost right, except the date is in the future.
Modified code:
final String domain = "http://www.twitter.com";
CookieSyncManager.createInstance(this);
CookieManager cookieManager = CookieManager.getInstance();
String cookiestring = cookieManager.getCookie(domain); //get all cookies
String[] cookies = cookiestring.split(";");
for (int i=0; i<cookies.length; i++) {
String[] cookieparts = cookies[i].split("="); //split cookie into name and value etc.
// set cookie to an expired date
cookieManager.setCookie(domain, cookieparts[0].trim()+"=; Expires=Wed, 31 Dec 2000 23:59:59 GMT");
}
CookieSyncManager.getInstance().sync(); //sync the new cookies just to be sure