The question is about linear combinations of real numbers, where some of the numbers are irrational, and can be interpreted as asking whether there are non-trivial linear relations (where trivial ones are things like $\pi + (5-\pi) = 5$, that are general algebraic properties of all numbers).
In general the expectation is that that given $k$ different irrational numbers, there will not be any non-trivial relations between them unless the numbers are constructed from fewer than $k$ "truly different" sources of irrationality, with some redundancy added. Relations can be interpreted as linear or polynomial (algebraic) relations with rational coefficients. There could, hypothetically, be suprises, such as nonzero integers $a,b,c$ (if any do exist) for which $a + b\pi + ce=0$, and there are a few known types of surprising relations, such as $r_1 = (\sum 1/n^2)$ being related to $r_2 = \pi^2$ by $6r_1 = r_2$. Nevertheless, in concrete situations there is usually at least a specific conjecture as to whether or not a relation with rational coefficients can exist between the numbers. If there is no apparent reason why a relationship should exist, it usually does not.
The formal description of these ideas is in terms of linear independence (over the rational numbers) and transcendence degree (also relative to the rational numbers) of a finite set of real numbers. The case where all the real numbers satisfy algebraic equations is covered by algebraic number theory and is well understood, as is the case where one number is transcendental. The case where several essentially different transcendental numbers are involved is the subject of transcendence theory, where there are many difficult conjectures and fewer theorems defining what is expected in most cases. A typical irrational-looking expression like $\pi + e^e$ has "probability 1" of being irrational but no known proof that it is irrational. A set of several irrationals such as $\pi$, $e$, $\zeta(5)$ will usually be as transcendental as possible (maximum transcendence degree) but the proof is out of reach.