The property in question was introduced by Ralph Ford in his dissertation, 1963. A space is totally paracompact if every open basis has a locally finite subcover.
The property may be further differentiated, depending on whether you start your definition with "every open basis $\mathcal B$" (which I assume you mean, and which coincides with total paracompactness), or you say
"there is an open basis $\mathcal B$, etc", which requires consideration of some other conditions to make the latter property different from paracompactness, given that the topology itself is a basis. One variation is base-paracompactness, when you say there is a base $\mathcal B$ of cardinality the weight of the space such that every open cover has a localy-finite refinement with elements of this base. Another variation is if one says there is a base $\mathcal B$ such that every base $\mathcal B'\subseteq\mathcal B$ has a locally-finite subcover, this is called base-base paracompactness.
The comment by bof above provides an example of a paracompact space that is not totally paracompact and answers your question. I will put together some links to the literature where you can read more.
H. H. Corson, T. J. McMinn, E. A. Michael, J. Nagata,
Bases and local finiteness.
Notices Amer. Math. Soc. 6 (1959), 814 (abstract).
One result announced there:
The base of all bounded, open,
convex sets in a reflexive, infinite-dimensional Banach space is coarse (i.e. it has no locally-finite subcover). Also the irrationals have a coarse base.
H. H. Corson, Collections of convex sets which cover a Banach space.
Fund. Math. 49 (1960/1961), 143–145.
This paper has a proof of the result announced earlier.
R. Ford, Basic properties in dimension theory,
Dissertation, Auburn Univ., (1963).
He proved that small and large inductive dimension coincide for totally paracompact metric spaces.
(the above is not a link to the dissertation itself, and the following is a link that I don't know if it works)
A.Lelek, On totally paracompact metric spaces.
Proc. Amer. Math. Soc. 19 (1968), 168–170
A. Lelek, Some cover properties of spaces
Fundamenta Mathematicae (1969)
Volume: 64, Issue: 2, page 209-218
One of the properties considered is total paracompactness.
Some methods of determining total paracompactness,
Dissertation, Auburn University (1982).
John O’Farrell, The Sorgenfrey line is not totally metacompact,
Houston J. Math 9 (1983), no.2, 271-273.
The irrationals with the usual topology are not totally paracompact. The Sorgenfrey line is not totally paracompact.
Zoltan Balogh and Harold Bennett
Total Paracompactness of Real GO-Spaces
Proceedings of the American Mathematical Society
Vol. 101, No. 4 (Dec., 1987), pp. 753-760
Real GO-spaces here means take the reals as a set, with a generalized-ordered topology: Any topology stronger than the usual, that has a basis of order-convex sets. Some examples are the Sorgenfrey line and the Michael line.
Francisco Gallego Lupiañez
Total paracompactness and Banach spaces
Proc. Amer. Math. Soc. 103 (1988), 210-214
(Seems very relevant, though I just discovered it with google.)
John E. Porter, Base-paracompact spaces
Topology and its Applications 128 (2003) 145–156
Base-paracompactness is a less restrictive property, there is a base $\mathcal B$ of cardinality of the weight, such that every open cover has a locally-finite refinement with elements of $\mathcal B$. At present 6/26/2019, it is an open question if every paracompact space is base-paracompact. The introduction lists some literature about total paracompactness.
Strashimir G. Popvassilev, Base-family paracompactness
Houston Journal of Mathematics
Volume 32, No. 2, 2006
Base-family paracompactness is a related property, and the introductions lists some of the results and literature relevant to total paracompactness.
Strashimir G. Popvassilev, Base-cover paracompactness,
Proc. Amer. Math. Soc. 132 (2004), 3121-3130
Another related property and discussion of literature.
Strashimir G. Popvassilev
Base-base paracompactness and subsets of the Sorgenfrey line
Mathematica Bohemica (2012)
Volume: 137, Issue: 4, page 395-401
Another related property and an open question.
Gary Gruenhage, Generalized metrizable spaces
In Recent progress of general topology III,
see section 20. Base paracompact