Short Answer: No. Relational databases are created out of the need to support certain business needs which NoSQL databases cannot replace. NoSQL databases addresses business use cases where relational databases fall short.
Long Answer: Now we are going to get in to the details:
RDBMS provides strong consistency guarantees: specifically, it guarantees ACID (Atomicity - all or nothing of the transaction is committed, Consistency - transactions leave the database in a consistent state - data stays valid in terms of integrity constraints, Isolation - each transaction is independent and Durability - once committed, the data stays ) properties. Due to the strong consistency guarantee, all users/apps see the same data at the same time. Downside of such strong guarantees is that they are more difficult to scale.
On the other hand, NoSQL systems support BASE (Basically Available - available as in CAP theorem, Soft state - state changes over time, Eventually consistent - with time all users will see the same output) properties which are less strict than ACID properties. Due to the loose consistency guarantee, NoSQL systems scale out easier than RDBMS. Further, unlike RDBMS systems, NoSQL systems do not have fixed schemas to work with. Developers decide the schema in their code which may lead to faster development.
As you can see, RDBMS and NoSQL systems demonstrate different set of properties. When you want to have fast data at the expense of providing inaccurate data at times (e.g. Twitter/FB feeds) and/or scale based on changing demand, NoSQL is a better solution. However, when you want to have super valid/consistent data and support complex and dynamic queries, RDBMS is a better solution. In fact, there are already attacks on NoSQL based systems due to lack of consistency. The recent attack on flexcoin is one such example. This teaches us to use the right tool for the job at hand. Hence, it is less like to see the demise of RDBMS technology in the near future.