If we are loading a closed source - dll dynamically in the source code which comes under three clause BSD license then is it mandatory to change license of my closed source dll to three clause BSD License? Also is it compulsory to publish this closed source dll code?
No, you do not need to change the license of your dll and you do not need to publish any source code. In contrast to a strong copyleft, non-permissive license like the GPL the BSD licenses are instead non-copyleft and permissive.
Copyleft basically means that any modifications you make to a piece of copyleft-licensed software have to be released under the same conditions as the original software. If you make a modification to a program licensed under the GPL the modification also needs to be GPL. See https://www.gnu.org/licenses/copyleft.html. This is not the case with the BSD license family.
Permissive is just what it sounds like. You're less restricted in what you are allowed to do with software under a what is considered a permissive open software license than what is allowed by a non-permissive license. You still have some obligations, such as attributing the authors, but a permissive license is generally easier to comply with than a non-permissive license.