Let $K=\mathbb{Q}_p$, $G=SL(2,K)$ and $P$ the subgroup of upper-triangular matrices. Let $V_P$ be a 1-dimensional representation of $P$ over $\mathbb{C}$ (i.e. a multiplicative character of $K$). Consider the induced representation $V_G$ as constructed by $$ V_G=\mathbb{C}[G]\otimes_{\mathbb{C}[P]}V_P. $$ As usual, this can be identified with $V_P$-valued functions on $G/P$ with an obvious $G$-action. I do not want to be specific about subtleties such as how precisely one should define $\mathbb{C}[G]$ for the infinite $G$ or which class of functions one should consider on $G/P$; results on any version of this construction are welcome.
I basically want to know if $V_G$ is irreducible. I am aware of a number of results which apply when $V_P$ is a unitary character, but I am specifically interested in non-unitary characters such as e.g. $x\to |x|_p^k$ for any $k\in\mathbb{R}$. Are there any known results on this more general case?
First of all, this will depend on what category of representations you want to work with: are you viewing $G$ as an algebraic group or as a locally profinite group? Since $|\cdot|_p$ isn't algebraic, I'm going to guess that you mean the latter.
There are a lot of subtleties that you need to take into account before this question can make sense. You need to focus on certain representations (smooth, admissible) and, as you point out, consider the appropriate replacement for $\Bbb{C}[G]$. In this case, it's the Hecke algebra of locally constant compactly supported functions $G\rightarrow\Bbb{C}$, under convolution. The first few sections of Bushnell--Henniart The local Langlands conjecture for $GL(2)$ give a good introduction to all of this.
In general, the question of irreducibility is quite difficult, if you genuinely want to consider any representation induced from a character of $P$. Usually one would only consider characters of the diagonal torus $T$ in $G$ which are inflated to $P$ through the quotient map $P\rightarrow T$, in which the classification is quite easy (although you should learn about $GL_2$ first, as it's easier, and the $SL_2$ classification is most easily stated in terms of that for $GL_2$. This is also in Bushnell--Henniart).