I wanted to add some things to the comments I had already made but the list of
comments have become very large and the comments I have already made are
becoming more and more difficult to follow so I'll put everything (including the
things I have already said) here instead even though it is not an answer to the
question.
Let us first consider the case of a minimal surface $X$ (by minimal I mean $K_X$
nef). Dolgachev (Dolgachev: Reflection groups in algebraic geometry is a good
reference even though the proof is only referenced there not given) gave a kind of
structure theorem for the image $A_X$ of $\mathrm{Aut}(X)$ in
$\mathrm{Aut}(S_X)$, where $S_X$ is the orthogonal complement of $K_X$ in
$\mathrm{NS}(X)$ modulo torsion. His result says that there is a normal subgroup
$W_X$ of $\mathrm{Aut}(S_X)$ generated by reflections in $-2$-curves and the
group $P_X$ generated by $A_X$ and $W_X$ is a semi-direct product and of finite
index in $\mathrm{Aut}(S_X)$. Note that it is possible to have $W_X=\{e\}$ and
then $A_X$ itself is of finite index and hence an arithmetic (and thus
finitely presented). It is also possible to have $W_X$ of finite index and then
$A_X$ is finite (and thus finitely presented). However, there are
intermediate cases where both $A_X$ and $W_X$ are infinite. Still $A_X$ is a
quotient of $P_X$ and hence is finitely generated. I do not know if it is always
finitely presented. Borcherds (Coxeter groups, Lorentzian lattices, and $K3$
surfaces. Internat. Math. Res. Notices 1998) gives examples where it is (and
where it is even nicer) but also examples where it is finitely generated but not
arithmetic.
[[Added]]
I now realise that finite presentation is always true: For that we
only need to show that $W_X$ is normally generated in $\mathrm{Aut}(X)$ by a
finite number of elements and for that it is enough to show the same thing for
$\mathrm{Aut}(S_X)$. We know that $W_X$ is generated by reflections in $-2$-elements.
There are however only a finite number of conjugacy classes $-2$-elements. For
this it is, by standard lattice theoretic arguments, enough to prove that there
are only a finite number of isomorphism classes of orthogonal
complements. However, the discriminant of such a complement is bounded in terms
of the rank and discriminant of $S_X$ and there are only a finite number of
forms of bounded rank and discriminant.
[[/Added]]
A further step would be to blow up points of $X$ (still assumed minimal). As
$X$ is the unique minimal model any automorphism of the blowing up is given by
an automorphism of $X$ that permutes the blown up points (and the subgroup
fixing the points is commensurable with the full automorphism group). In
the case of abelian or hyperelliptic surfaces blowing up just one point is
pointless as it just serves to kill off the connected component of
$\mathrm{Aut}(X)$ so in that case the first interesting case is blowing up two
points.
Consider the case of blowing up two points when $X$ is abelian. So we have two
points on $X$ one of which we can assume is $0$ and the other we'll call $x$. An
automorphism of $X$ that fixes both of these points will be en automorphism of
$X$ as abelian variety that fixes pointwise the closed subgroup $A$ generated by
$x$. The group fixing $x$ will then have finite index in the the group fixing
$A$ pointwise. For any abelian subvariety $A$ of $X$, the subgroup of
$\mathrm{Aut}(X)$ fixing all the points of $A$ is an arithmetic subgroup (in a
not necessarily semi-simple group) and in particular is finitely presented.
The same argument works for abelian varieties of any dimension. There one of
course also has the option of blowing up positive dimensional varieties, assume
$S$ is a smooth closed subvariety. This
time the automorphism group is the subgroup of automorphisms $X$ that fixes
$S$. We thus get an induced action on $S$ and the kernel of that action has the
same structure as before. Unless I am mistaken, the automorphisms of $S$ that
extend to $X$ are of finite index in $\mathrm{Aut}(S)$ (look at $\mathrm{Alb}(S)
\rightarrow X$ and split it up to isogeny). Hence the finite generation etc for
the blowing up is reduced to finite generation for $S$ (and conversely for $X$
replaced by $\mathrm{Alb}(S)$).
Consider now the case of $X$ still minimal but non-abelian or hyper-elliptic and
look at blowing up of one point $x$. For a general point of $X$ (in the sense of
being outside a countable number of proper subvarieties) the automorphism group
is trivial and hence finitely generated. The situation for arbitrary $x$ seems
unclear but one thread of the discussion started concerning itself with whether
for a general $X$ there is a characterisation (up to commensurability) of
$\mathrm{Aut}(X)$ similar to the minimal case: Look at all automorphisms of the
integral cohomology of $X$ that preserves multiplicative structure, Hodge
structure, Chern classes (of the tangent bundle) and effective cones (spanned by
effective cycles). Is the image of the automorphism group of $X$ of finite index
in this group? I think the answer is no (and I hope that what I present here is
a proof). For that we need to recall some facts on Seshadri constants
(Lazarsfeld: Positivity in Algebraic Geometry, I is my reference). Given a point
$x$ the Seshadri constants $\epsilon(L;x)$ for $L$ nef (but also for $L$
restricted to be ample) determine (and are determined by) the nef cone of the
blowing up at $x$; $L-rE$ is nef precisely when $0\leq r\leq \epsilon(L;x)$.
Switching tack, there is a subset $U$ of $X$ which is the intersection of a
countable number of open non-empty subsets of $X$ such that $\epsilon(L;x)$ is
constant on $U$ for all ample $L$. Indeed, $\epsilon(L;x)$ can be expressed
(loc. cit.: 5.1.17) in terms of whether or not $kL$ separates $s$-jets at $x$
and for fixed $k$ and $s$ the separation is true on an open subset.
The conclusion is that there is a $U$ which is the intersection of a countable
number of non-empty open subsets for which the nef cone of the blowing up of
$X$ at $x$ is independent of $x$ when one expresses it in the decomposition
$\mathrm{NS}(X)\bigoplus\mathrm Z E$. If we assume now that $K_X$ is numerically
trivial we have that the first Chern class of the tangent bundle of the blowup
of $X$ at some $x$ equals $E$ (up to torsion) and hence the group above will
preserve the decomposition $\mathrm{NS}(X)\bigoplus\mathrm Z E$ and fix $E$ so
come from an automorphism of $\mathrm{NS}(X)$. The only further condition we put
on it is that it preserve the nef cone but for $x\in U$ this cone is independent
of $x$. As we can further arrange it so that $x\in U \implies \varphi(x)\in U$
(as $\mathrm{Aut}(X)$ is countable) we get that all elements of
$\mathrm{Aut}(X)$ give structure preserving automorphisms of the cohomology of
the blowup of $X$ at $x$. However, as observed before, at the price of shrinking
$U$ we can assume that that the automorphism group of the blowing up is
trivial. Hence, if we let $X$ be for instance a K3-surface with infinite
automorphism group we get an example.
Best Answer
Of the commenters on this question, two are authors (with Harald Helfgott) of the very nice paper "Root numbers and ranks in positive characteristic", which gives an example (under parity conjecture) of a non-isotrivial 1-parameter family of elliptic curves over a global function field $K = \mathbf{F}_q(u)$ (any odd $q$) such that each fiber $E_t$ for $t \in K$ has rank strictly greater than that of the generic fiber. This is conjectured to be impossible in the number field case.