Here is a solution inspired by Losif Pinelis, but addresses the existence of non-singular matrices (see comment to last answer).

First consider the case where diag(R)= 1_N, the "all ones vector", and |v_1| > |v_2| > ... > |v_N|.  Let {e1,...,eN} be the natural basis, q_i= v_i/v_1 e1 + \sqrt(1-(v_i/v_1)^2) e_i, 1<=i<=N, Qo= [q_1, q_2, ..., q_N], R= Qo'Qo.  Note that by construction ||q_i||= 1, Qo is lower triangular with positive diagonal entries, and thus is non-singular.  Also note, e_i'Qoe_1= v_i/v_1, so inv(Qo')v= v_1e_1, and thus, v'inv(R)v= v'inv(Qo)inv(Qo')v= ||inv(Qo')v||^2= ||v_1e_1||^2= v_1^2= ||v||_\Inf^2.  But given any R with diag(R)=1_N, there exists a Cholesky factorization, R= Q'*Q, and by the Cauchy-Schwartz Lemma, v'inv(R)v= ||inv(Q')v||^2 >= (q_1'inv(Q')v)^2=(e_1'Q inv(Q')v)^2= v_1^2= ||v||_\Inf^2; so, Ro is a global minimum; and min v'*inv(R)v= ||v||_\Inf^2.

Now consider the case where diag(R)= 1_N, and v sorted, by v's entries are not distinct.  The Qo as constructed above is singular, but since v'*inv(R)v is a continuous function of R, it can be extended to inf{v'*inv(R)v}= ||v||_\Inf^2.

Finally, consider the more general case.  Let \Lambda= diag(r)^(-1/2), and w= \Lambda v.  Also, for each R, let G= \Lambda R \Lambda, so diag(G)= 1_N.  Then, inf{v'*inv(R)v}= inf{v'\Lambda inv(\Lambda)inv(R) inv(\Lambda)\Lamba v}= inf{w'inv(G)w}= ||w||_\Inf^2= = ||diag(r)^(-1/2)v||_\Inf^2.  This proves inf{v'*inv(R)v}= ||diag(r)^(-1/2)v||_\Inf^2.

Now consider the second problem.  Let w= diag(r)^(1/2)sign(v), and Ro= ww'.  So, v'Ro v= sum{v_i v_j R[i,j]}= sum{v_i v_j \sqrt(r_i) sign(v_i) \sqrt(r_j) sign(v_j)}= ||diag(r)^(1/2)v||_1^2.  But |sum{v_i v_j R[i,j]}|<= sum{|v_i v_j R[i,j]|}<=sum{|v_i v_j \sqrt(r_i) \sqrt(r_j)|}= ||diag(r)^(1/2)v||_1^2; so, Ro is a global maximizer.  This proves sup{v'Rv}= ||diag(r)^(1/2)v||_1^2.  

Juxtaposing our solutions for parts 1 and 2, we have the following.
inf{v'*inv(R)v}= ||diag(r)^(-1/2)v||_\Inf^2
sup{v'Rv}= ||diag(r)^(1/2)v||_1^2
which exhibits a shocking duality.  Is this a coincidence, or a hint at something deeper?