Believe it or not, there is a chimney underneath all this ivy.  The homeowner called because he was having a draft issue. The ivy is the reason why.  Draft issues are one problem caused by ivy on a chimney.  The ivy also makes a great nesting place for birds or other critters, which may wind up in the chimney, causing an obstruction.  Ivy also damages the masonry.  It can get embedded in the mortar joints causing weakness in the mortar joints and can ultimately damage them.  Bottom line, even though it may look pretty, don't let the ivy grow up the chimney.  If you choose to allow it to grow, have the chimney inspected to be sure all is well with your chimney inside and out.