No, the boost/poco license does not require you to opensource your application.
Here' the relevant part from the boost license FAQ
How is the Boost license different from the GNU General Public License (GPL)? The Boost license permits the creation of derivative works for commercial or non-commercial use with no legal requirement to release your source code. Other differences include Boost not requiring reproduction of copyright messages for object code redistribution, and the fact that the Boost license is not "viral": if you distribute your own code along with some Boost code, the Boost license applies only to the Boost code (and modified versions thereof); you are free to license your own code under any terms you like. The GPL is also much longer, and thus may be harder to understand.